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I have seen this question asked a lot but never seen a true concrete answer to it. So I am going to post one here which will hopefully help people understand why exactly there is "modulo bias" when using a random number generator, like rand() in C++.
So rand() is a pseudo-random number generator which chooses a natural number between 0 and RAND_MAX, which is a constant defined in cstdlib (see this article for a general overview on rand()).
Now what happens if you want to generate a random number between say 0 and 2? For the sake of explanation, let's say RAND_MAX is 10 and I decide to generate a random number between 0 and 2 by calling rand()%3. However, rand()%3 does not produce the numbers between 0 and 2 with equal probability!
When rand() returns 0, 3, 6, or 9, rand()%3 == 0. Therefore, P(0) = 4/11
When rand() returns 1, 4, 7, or 10, rand()%3 == 1. Therefore, P(1) = 4/11
When rand() returns 2, 5, or 8, rand()%3 == 2. Therefore, P(2) = 3/11
This does not generate the numbers between 0 and 2 with equal probability. Of course for small ranges this might not be the biggest issue but for a larger range this could skew the distribution, biasing the smaller numbers.
So when does rand()%n return a range of numbers from 0 to n-1 with equal probability? When RAND_MAX%n == n - 1. In this case, along with our earlier assumption rand() does return a number between 0 and RAND_MAX with equal probability, the modulo classes of n would also be equally distributed.
So how do we solve this problem? A crude way is to keep generating random numbers until you get a number in your desired range:
int x;
do {
x = rand();
} while (x >= n);
but that's inefficient for low values of n, since you only have a n/RAND_MAX chance of getting a value in your range, and so you'll need to perform RAND_MAX/n calls to rand() on average.
A more efficient formula approach would be to take some large range with a length divisible by n, like RAND_MAX - RAND_MAX % n, keep generating random numbers until you get one that lies in the range, and then take the modulus:
int x;
do {
x = rand();
} while (x >= (RAND_MAX - RAND_MAX % n));
x %= n;
For small values of n, this will rarely require more than one call to rand().
Works cited and further reading:
CPlusPlus Reference
Eternally Confuzzled
Keep selecting a random is a good way to remove the bias.
Update
We could make the code fast if we search for an x in range divisible by n.
// Assumptions
// rand() in [0, RAND_MAX]
// n in (0, RAND_MAX]
int x;
// Keep searching for an x in a range divisible by n
do {
x = rand();
} while (x >= RAND_MAX - (RAND_MAX % n))
x %= n;
The above loop should be very fast, say 1 iteration on average.
#user1413793 is correct about the problem. I'm not going to discuss that further, except to make one point: yes, for small values of n and large values of RAND_MAX, the modulo bias can be very small. But using a bias-inducing pattern means that you must consider the bias every time you calculate a random number and choose different patterns for different cases. And if you make the wrong choice, the bugs it introduces are subtle and almost impossible to unit test. Compared to just using the proper tool (such as arc4random_uniform), that's extra work, not less work. Doing more work and getting a worse solution is terrible engineering, especially when doing it right every time is easy on most platforms.
Unfortunately, the implementations of the solution are all incorrect or less efficient than they should be. (Each solution has various comments explaining the problems, but none of the solutions have been fixed to address them.) This is likely to confuse the casual answer-seeker, so I'm providing a known-good implementation here.
Again, the best solution is just to use arc4random_uniform on platforms that provide it, or a similar ranged solution for your platform (such as Random.nextInt on Java). It will do the right thing at no code cost to you. This is almost always the correct call to make.
If you don't have arc4random_uniform, then you can use the power of opensource to see exactly how it is implemented on top of a wider-range RNG (ar4random in this case, but a similar approach could also work on top of other RNGs).
Here is the OpenBSD implementation:
/*
* Calculate a uniformly distributed random number less than upper_bound
* avoiding "modulo bias".
*
* Uniformity is achieved by generating new random numbers until the one
* returned is outside the range [0, 2**32 % upper_bound). This
* guarantees the selected random number will be inside
* [2**32 % upper_bound, 2**32) which maps back to [0, upper_bound)
* after reduction modulo upper_bound.
*/
u_int32_t
arc4random_uniform(u_int32_t upper_bound)
{
u_int32_t r, min;
if (upper_bound < 2)
return 0;
/* 2**32 % x == (2**32 - x) % x */
min = -upper_bound % upper_bound;
/*
* This could theoretically loop forever but each retry has
* p > 0.5 (worst case, usually far better) of selecting a
* number inside the range we need, so it should rarely need
* to re-roll.
*/
for (;;) {
r = arc4random();
if (r >= min)
break;
}
return r % upper_bound;
}
It is worth noting the latest commit comment on this code for those who need to implement similar things:
Change arc4random_uniform() to calculate 2**32 % upper_bound as
-upper_bound % upper_bound. Simplifies the code and makes it the
same on both ILP32 and LP64 architectures, and also slightly faster on
LP64 architectures by using a 32-bit remainder instead of a 64-bit
remainder.
Pointed out by Jorden Verwer on tech#
ok deraadt; no objections from djm or otto
The Java implementation is also easily findable (see previous link):
public int nextInt(int n) {
if (n <= 0)
throw new IllegalArgumentException("n must be positive");
if ((n & -n) == n) // i.e., n is a power of 2
return (int)((n * (long)next(31)) >> 31);
int bits, val;
do {
bits = next(31);
val = bits % n;
} while (bits - val + (n-1) < 0);
return val;
}
Definition
Modulo Bias is the inherent bias in using modulo arithmetic to reduce an output set to a subset of the input set. In general, a bias exists whenever the mapping between the input and output set is not equally distributed, as in the case of using modulo arithmetic when the size of the output set is not a divisor of the size of the input set.
This bias is particularly hard to avoid in computing, where numbers are represented as strings of bits: 0s and 1s. Finding truly random sources of randomness is also extremely difficult, but is beyond the scope of this discussion. For the remainder of this answer, assume that there exists an unlimited source of truly random bits.
Problem Example
Let's consider simulating a die roll (0 to 5) using these random bits. There are 6 possibilities, so we need enough bits to represent the number 6, which is 3 bits. Unfortunately, 3 random bits yields 8 possible outcomes:
000 = 0, 001 = 1, 010 = 2, 011 = 3
100 = 4, 101 = 5, 110 = 6, 111 = 7
We can reduce the size of the outcome set to exactly 6 by taking the value modulo 6, however this presents the modulo bias problem: 110 yields a 0, and 111 yields a 1. This die is loaded.
Potential Solutions
Approach 0:
Rather than rely on random bits, in theory one could hire a small army to roll dice all day and record the results in a database, and then use each result only once. This is about as practical as it sounds, and more than likely would not yield truly random results anyway (pun intended).
Approach 1:
Instead of using the modulus, a naive but mathematically correct solution is to discard results that yield 110 and 111 and simply try again with 3 new bits. Unfortunately, this means that there is a 25% chance on each roll that a re-roll will be required, including each of the re-rolls themselves. This is clearly impractical for all but the most trivial of uses.
Approach 2:
Use more bits: instead of 3 bits, use 4. This yield 16 possible outcomes. Of course, re-rolling anytime the result is greater than 5 makes things worse (10/16 = 62.5%) so that alone won't help.
Notice that 2 * 6 = 12 < 16, so we can safely take any outcome less than 12 and reduce that modulo 6 to evenly distribute the outcomes. The other 4 outcomes must be discarded, and then re-rolled as in the previous approach.
Sounds good at first, but let's check the math:
4 discarded results / 16 possibilities = 25%
In this case, 1 extra bit didn't help at all!
That result is unfortunate, but let's try again with 5 bits:
32 % 6 = 2 discarded results; and
2 discarded results / 32 possibilities = 6.25%
A definite improvement, but not good enough in many practical cases. The good news is, adding more bits will never increase the chances of needing to discard and re-roll. This holds not just for dice, but in all cases.
As demonstrated however, adding an 1 extra bit may not change anything. In fact if we increase our roll to 6 bits, the probability remains 6.25%.
This begs 2 additional questions:
If we add enough bits, is there a guarantee that the probability of a discard will diminish?
How many bits are enough in the general case?
General Solution
Thankfully the answer to the first question is yes. The problem with 6 is that 2^x mod 6 flips between 2 and 4 which coincidentally are a multiple of 2 from each other, so that for an even x > 1,
[2^x mod 6] / 2^x == [2^(x+1) mod 6] / 2^(x+1)
Thus 6 is an exception rather than the rule. It is possible to find larger moduli that yield consecutive powers of 2 in the same way, but eventually this must wrap around, and the probability of a discard will be reduced.
Without offering further proof, in general using double the number
of bits required will provide a smaller, usually insignificant,
chance of a discard.
Proof of Concept
Here is an example program that uses OpenSSL's libcrypo to supply random bytes. When compiling, be sure to link to the library with -lcrypto which most everyone should have available.
#include <iostream>
#include <assert.h>
#include <limits>
#include <openssl/rand.h>
volatile uint32_t dummy;
uint64_t discardCount;
uint32_t uniformRandomUint32(uint32_t upperBound)
{
assert(RAND_status() == 1);
uint64_t discard = (std::numeric_limits<uint64_t>::max() - upperBound) % upperBound;
RAND_bytes((uint8_t*)(&randomPool), sizeof(randomPool));
while(randomPool > (std::numeric_limits<uint64_t>::max() - discard)) {
RAND_bytes((uint8_t*)(&randomPool), sizeof(randomPool));
++discardCount;
}
return randomPool % upperBound;
}
int main() {
discardCount = 0;
const uint32_t MODULUS = (1ul << 31)-1;
const uint32_t ROLLS = 10000000;
for(uint32_t i = 0; i < ROLLS; ++i) {
dummy = uniformRandomUint32(MODULUS);
}
std::cout << "Discard count = " << discardCount << std::endl;
}
I encourage playing with the MODULUS and ROLLS values to see how many re-rolls actually happen under most conditions. A sceptical person may also wish to save the computed values to file and verify the distribution appears normal.
Mark's Solution (The accepted solution) is Nearly Perfect.
int x;
do {
x = rand();
} while (x >= (RAND_MAX - RAND_MAX % n));
x %= n;
edited Mar 25 '16 at 23:16
Mark Amery 39k21170211
However, it has a caveat which discards 1 valid set of outcomes in any scenario where RAND_MAX (RM) is 1 less than a multiple of N (Where N = the Number of possible valid outcomes).
ie, When the 'count of values discarded' (D) is equal to N, then they are actually a valid set (V), not an invalid set (I).
What causes this is at some point Mark loses sight of the difference between N and Rand_Max.
N is a set who's valid members are comprised only of Positive Integers, as it contains a count of responses that would be valid. (eg: Set N = {1, 2, 3, ... n } )
Rand_max However is a set which ( as defined for our purposes ) includes any number of non-negative integers.
In it's most generic form, what is defined here as Rand Max is the Set of all valid outcomes, which could theoretically include negative numbers or non-numeric values.
Therefore Rand_Max is better defined as the set of "Possible Responses".
However N operates against the count of the values within the set of valid responses, so even as defined in our specific case, Rand_Max will be a value one less than the total number it contains.
Using Mark's Solution, Values are Discarded when: X => RM - RM % N
EG:
Ran Max Value (RM) = 255
Valid Outcome (N) = 4
When X => 252, Discarded values for X are: 252, 253, 254, 255
So, if Random Value Selected (X) = {252, 253, 254, 255}
Number of discarded Values (I) = RM % N + 1 == N
IE:
I = RM % N + 1
I = 255 % 4 + 1
I = 3 + 1
I = 4
X => ( RM - RM % N )
255 => (255 - 255 % 4)
255 => (255 - 3)
255 => (252)
Discard Returns $True
As you can see in the example above, when the value of X (the random number we get from the initial function) is 252, 253, 254, or 255 we would discard it even though these four values comprise a valid set of returned values.
IE: When the count of the values Discarded (I) = N (The number of valid outcomes) then a Valid set of return values will be discarded by the original function.
If we describe the difference between the values N and RM as D, ie:
D = (RM - N)
Then as the value of D becomes smaller, the Percentage of unneeded re-rolls due to this method increases at each natural multiplicative. (When RAND_MAX is NOT equal to a Prime Number this is of valid concern)
EG:
RM=255 , N=2 Then: D = 253, Lost percentage = 0.78125%
RM=255 , N=4 Then: D = 251, Lost percentage = 1.5625%
RM=255 , N=8 Then: D = 247, Lost percentage = 3.125%
RM=255 , N=16 Then: D = 239, Lost percentage = 6.25%
RM=255 , N=32 Then: D = 223, Lost percentage = 12.5%
RM=255 , N=64 Then: D = 191, Lost percentage = 25%
RM=255 , N= 128 Then D = 127, Lost percentage = 50%
Since the percentage of Rerolls needed increases the closer N comes to RM, this can be of valid concern at many different values depending on the constraints of the system running he code and the values being looked for.
To negate this we can make a simple amendment As shown here:
int x;
do {
x = rand();
} while (x > (RAND_MAX - ( ( ( RAND_MAX % n ) + 1 ) % n) );
x %= n;
This provides a more general version of the formula which accounts for the additional peculiarities of using modulus to define your max values.
Examples of using a small value for RAND_MAX which is a multiplicative of N.
Mark'original Version:
RAND_MAX = 3, n = 2, Values in RAND_MAX = 0,1,2,3, Valid Sets = 0,1 and 2,3.
When X >= (RAND_MAX - ( RAND_MAX % n ) )
When X >= 2 the value will be discarded, even though the set is valid.
Generalized Version 1:
RAND_MAX = 3, n = 2, Values in RAND_MAX = 0,1,2,3, Valid Sets = 0,1 and 2,3.
When X > (RAND_MAX - ( ( RAND_MAX % n ) + 1 ) % n )
When X > 3 the value would be discarded, but this is not a vlue in the set RAND_MAX so there will be no discard.
Additionally, in the case where N should be the number of values in RAND_MAX; in this case, you could set N = RAND_MAX +1, unless RAND_MAX = INT_MAX.
Loop-wise you could just use N = 1, and any value of X will be accepted, however, and put an IF statement in for your final multiplier. But perhaps you have code that may have a valid reason to return a 1 when the function is called with n = 1...
So it may be better to use 0, which would normally provide a Div 0 Error, when you wish to have n = RAND_MAX+1
Generalized Version 2:
int x;
if n != 0 {
do {
x = rand();
} while (x > (RAND_MAX - ( ( ( RAND_MAX % n ) + 1 ) % n) );
x %= n;
} else {
x = rand();
}
Both of these solutions resolve the issue with needlessly discarded valid results which will occur when RM+1 is a product of n.
The second version also covers the edge case scenario when you need n to equal the total possible set of values contained in RAND_MAX.
The modified approach in both is the same and allows for a more general solution to the need of providing valid random numbers and minimizing discarded values.
To reiterate:
The Basic General Solution which extends mark's example:
// Assumes:
// RAND_MAX is a globally defined constant, returned from the environment.
// int n; // User input, or externally defined, number of valid choices.
int x;
do {
x = rand();
} while (x > (RAND_MAX - ( ( ( RAND_MAX % n ) + 1 ) % n) ) );
x %= n;
The Extended General Solution which Allows one additional scenario of RAND_MAX+1 = n:
// Assumes:
// RAND_MAX is a globally defined constant, returned from the environment.
// int n; // User input, or externally defined, number of valid choices.
int x;
if n != 0 {
do {
x = rand();
} while (x > (RAND_MAX - ( ( ( RAND_MAX % n ) + 1 ) % n) ) );
x %= n;
} else {
x = rand();
}
In some languages ( particularly interpreted languages ) doing the calculations of the compare-operation outside of the while condition may lead to faster results as this is a one-time calculation no matter how many re-tries are required. YMMV!
// Assumes:
// RAND_MAX is a globally defined constant, returned from the environment.
// int n; // User input, or externally defined, number of valid choices.
int x; // Resulting random number
int y; // One-time calculation of the compare value for x
y = RAND_MAX - ( ( ( RAND_MAX % n ) + 1 ) % n)
if n != 0 {
do {
x = rand();
} while (x > y);
x %= n;
} else {
x = rand();
}
There are two usual complaints with the use of modulo.
one is valid for all generators. It is easier to see in a limit case. If your generator has a RAND_MAX which is 2 (that isn't compliant with the C standard) and you want only 0 or 1 as value, using modulo will generate 0 twice as often (when the generator generates 0 and 2) as it will generate 1 (when the generator generates 1). Note that this is true as soon as you don't drop values, whatever the mapping you are using from the generator values to the wanted one, one will occurs twice as often as the other.
some kind of generator have their less significant bits less random than the other, at least for some of their parameters, but sadly those parameter have other interesting characteristic (such has being able to have RAND_MAX one less than a power of 2). The problem is well known and for a long time library implementation probably avoid the problem (for instance the sample rand() implementation in the C standard use this kind of generator, but drop the 16 less significant bits), but some like to complain about that and you may have bad luck
Using something like
int alea(int n){
assert (0 < n && n <= RAND_MAX);
int partSize =
n == RAND_MAX ? 1 : 1 + (RAND_MAX-n)/(n+1);
int maxUsefull = partSize * n + (partSize-1);
int draw;
do {
draw = rand();
} while (draw > maxUsefull);
return draw/partSize;
}
to generate a random number between 0 and n will avoid both problems (and it avoids overflow with RAND_MAX == INT_MAX)
BTW, C++11 introduced standard ways to the the reduction and other generator than rand().
With a RAND_MAX value of 3 (in reality it should be much higher than that but the bias would still exist) it makes sense from these calculations that there is a bias:
1 % 2 = 1
2 % 2 = 0
3 % 2 = 1
random_between(1, 3) % 2 = more likely a 1
In this case, the % 2 is what you shouldn't do when you want a random number between 0 and 1. You could get a random number between 0 and 2 by doing % 3 though, because in this case: RAND_MAX is a multiple of 3.
Another method
There is much simpler but to add to other answers, here is my solution to get a random number between 0 and n - 1, so n different possibilities, without bias.
the number of bits (not bytes) needed to encode the number of possibilities is the number of bits of random data you'll need
encode the number from random bits
if this number is >= n, restart (no modulo).
Really random data is not easy to obtain, so why use more bits than needed.
Below is an example in Smalltalk, using a cache of bits from a pseudo-random number generator. I'm no security expert so use at your own risk.
next: n
| bitSize r from to |
n < 0 ifTrue: [^0 - (self next: 0 - n)].
n = 0 ifTrue: [^nil].
n = 1 ifTrue: [^0].
cache isNil ifTrue: [cache := OrderedCollection new].
cache size < (self randmax highBit) ifTrue: [
Security.DSSRandom default next asByteArray do: [ :byte |
(1 to: 8) do: [ :i | cache add: (byte bitAt: i)]
]
].
r := 0.
bitSize := n highBit.
to := cache size.
from := to - bitSize + 1.
(from to: to) do: [ :i |
r := r bitAt: i - from + 1 put: (cache at: i)
].
cache removeFrom: from to: to.
r >= n ifTrue: [^self next: n].
^r
Modulo reduction is a commonly seen way to make a random integer generator avoid the worst case of running forever.
When the range of possible integers is unknown, however, there is no way in general to "fix" this worst case of running forever without introducing bias. It's not just modulo reduction (rand() % n, discussed in the accepted answer) that will introduce bias this way, but also the "multiply-and-shift" reduction of Daniel Lemire, or if you stop rejecting an outcome after a set number of iterations. (To be clear, this doesn't mean there is no way to fix the bias issues present in pseudorandom generators. For example, even though modulo and other reductions are biased in general, they will have no issues with bias if the range of possible integers is a power of 2 and if the random generator produces unbiased random bits or blocks of them.)
The following answer of mine discusses the relationship between running time and bias in random generators, assuming we have a "true" random generator that can produce unbiased and independent random bits. The answer doesn't even involve the rand() function in C because it has many issues. Perhaps the most serious here is the fact that the C standard does not explicitly specify a particular distribution for the numbers returned by rand(), not even a uniform distribution.
How to generate a random integer in the range [0,n] from a stream of random bits without wasting bits?
As the accepted answer indicates, "modulo bias" has its roots in the low value of RAND_MAX. He uses an extremely small value of RAND_MAX (10) to show that if RAND_MAX were 10, then you tried to generate a number between 0 and 2 using %, the following outcomes would result:
rand() % 3 // if RAND_MAX were only 10, gives
output of rand() | rand()%3
0 | 0
1 | 1
2 | 2
3 | 0
4 | 1
5 | 2
6 | 0
7 | 1
8 | 2
9 | 0
So there are 4 outputs of 0's (4/10 chance) and only 3 outputs of 1 and 2 (3/10 chances each).
So it's biased. The lower numbers have a better chance of coming out.
But that only shows up so obviously when RAND_MAX is small. Or more specifically, when the number your are modding by is large compared to RAND_MAX.
A much better solution than looping (which is insanely inefficient and shouldn't even be suggested) is to use a PRNG with a much larger output range. The Mersenne Twister algorithm has a maximum output of 4,294,967,295. As such doing MersenneTwister::genrand_int32() % 10 for all intents and purposes, will be equally distributed and the modulo bias effect will all but disappear.
I just wrote a code for Von Neumann's Unbiased Coin Flip Method, that should theoretically eliminate any bias in the random number generation process. More info can be found at (http://en.wikipedia.org/wiki/Fair_coin)
int unbiased_random_bit() {
int x1, x2, prev;
prev = 2;
x1 = rand() % 2;
x2 = rand() % 2;
for (;; x1 = rand() % 2, x2 = rand() % 2)
{
if (x1 ^ x2) // 01 -> 1, or 10 -> 0.
{
return x2;
}
else if (x1 & x2)
{
if (!prev) // 0011
return 1;
else
prev = 1; // 1111 -> continue, bias unresolved
}
else
{
if (prev == 1)// 1100
return 0;
else // 0000 -> continue, bias unresolved
prev = 0;
}
}
}
I had this question on an Algorithms test yesterday, and I can't figure out the answer. It is driving me absolutely crazy, because it was worth about 40 points. I figure that most of the class didn't solve it correctly, because I haven't come up with a solution in the past 24 hours.
Given a arbitrary binary string of length n, find three evenly spaced ones within the string if they exist. Write an algorithm which solves this in O(n * log(n)) time.
So strings like these have three ones that are "evenly spaced": 11100000, 0100100100
edit: It is a random number, so it should be able to work for any number. The examples I gave were to illustrate the "evenly spaced" property. So 1001011 is a valid number. With 1, 4, and 7 being ones that are evenly spaced.
Finally! Following up leads in sdcvvc's answer, we have it: the O(n log n) algorithm for the problem! It is simple too, after you understand it. Those who guessed FFT were right.
The problem: we are given a binary string S of length n, and we want to find three evenly spaced 1s in it. For example, S may be 110110010, where n=9. It has evenly spaced 1s at positions 2, 5, and 8.
Scan S left to right, and make a list L of positions of 1. For the S=110110010 above, we have the list L = [1, 2, 4, 5, 8]. This step is O(n). The problem is now to find an arithmetic progression of length 3 in L, i.e. to find distinct a, b, c in L such that b-a = c-b, or equivalently a+c=2b. For the example above, we want to find the progression (2, 5, 8).
Make a polynomial p with terms xk for each k in L. For the example above, we make the polynomial p(x) = (x + x2 + x4 + x5+x8). This step is O(n).
Find the polynomial q = p2, using the Fast Fourier Transform. For the example above, we get the polynomial q(x) = x16 + 2x13 + 2x12 + 3x10 + 4x9 + x8 + 2x7 + 4x6 + 2x5 + x4 + 2x3 + x2. This step is O(n log n).
Ignore all terms except those corresponding to x2k for some k in L. For the example above, we get the terms x16, 3x10, x8, x4, x2. This step is O(n), if you choose to do it at all.
Here's the crucial point: the coefficient of any x2b for b in L is precisely the number of pairs (a,c) in L such that a+c=2b. [CLRS, Ex. 30.1-7] One such pair is (b,b) always (so the coefficient is at least 1), but if there exists any other pair (a,c), then the coefficient is at least 3, from (a,c) and (c,a). For the example above, we have the coefficient of x10 to be 3 precisely because of the AP (2,5,8). (These coefficients x2b will always be odd numbers, for the reasons above. And all other coefficients in q will always be even.)
So then, the algorithm is to look at the coefficients of these terms x2b, and see if any of them is greater than 1. If there is none, then there are no evenly spaced 1s. If there is a b in L for which the coefficient of x2b is greater than 1, then we know that there is some pair (a,c) — other than (b,b) — for which a+c=2b. To find the actual pair, we simply try each a in L (the corresponding c would be 2b-a) and see if there is a 1 at position 2b-a in S. This step is O(n).
That's all, folks.
One might ask: do we need to use FFT? Many answers, such as beta's, flybywire's, and rsp's, suggest that the approach that checks each pair of 1s and sees if there is a 1 at the "third" position, might work in O(n log n), based on the intuition that if there are too many 1s, we would find a triple easily, and if there are too few 1s, checking all pairs takes little time. Unfortunately, while this intuition is correct and the simple approach is better than O(n2), it is not significantly better. As in sdcvvc's answer, we can take the "Cantor-like set" of strings of length n=3k, with 1s at the positions whose ternary representation has only 0s and 2s (no 1s) in it. Such a string has 2k = n(log 2)/(log 3) ≈ n0.63 ones in it and no evenly spaced 1s, so checking all pairs would be of the order of the square of the number of 1s in it: that's 4k ≈ n1.26 which unfortunately is asymptotically much larger than (n log n). In fact, the worst case is even worse: Leo Moser in 1953 constructed (effectively) such strings which have n1-c/√(log n) 1s in them but no evenly spaced 1s, which means that on such strings, the simple approach would take Θ(n2-2c/√(log n)) — only a tiny bit better than Θ(n2), surprisingly!
About the maximum number of 1s in a string of length n with no 3 evenly spaced ones (which we saw above was at least n0.63 from the easy Cantor-like construction, and at least n1-c/√(log n) with Moser's construction) — this is OEIS A003002. It can also be calculated directly from OEIS A065825 as the k such that A065825(k) ≤ n < A065825(k+1). I wrote a program to find these, and it turns out that the greedy algorithm does not give the longest such string. For example, for n=9, we can get 5 1s (110100011) but the greedy gives only 4 (110110000), for n=26 we can get 11 1s (11001010001000010110001101) but the greedy gives only 8 (11011000011011000000000000), and for n=74 we can get 22 1s (11000010110001000001011010001000000000000000010001011010000010001101000011) but the greedy gives only 16 (11011000011011000000000000011011000011011000000000000000000000000000000000). They do agree at quite a few places until 50 (e.g. all of 38 to 50), though. As the OEIS references say, it seems that Jaroslaw Wroblewski is interested in this question, and he maintains a website on these non-averaging sets. The exact numbers are known only up to 194.
Your problem is called AVERAGE in this paper (1999):
A problem is 3SUM-hard if there is a sub-quadratic reduction from the problem 3SUM: Given a set A of n integers, are there elements a,b,c in A such that a+b+c = 0? It is not known whether AVERAGE is 3SUM-hard. However, there is a simple linear-time reduction from AVERAGE to 3SUM, whose description we omit.
Wikipedia:
When the integers are in the range [−u ... u], 3SUM can be solved in time O(n + u lg u) by representing S as a bit vector and performing a convolution using FFT.
This is enough to solve your problem :).
What is very important is that O(n log n) is complexity in terms of number of zeroes and ones, not the count of ones (which could be given as an array, like [1,5,9,15]). Checking if a set has an arithmetic progression, terms of number of 1's, is hard, and according to that paper as of 1999 no faster algorithm than O(n2) is known, and is conjectured that it doesn't exist. Everybody who doesn't take this into account is attempting to solve an open problem.
Other interesting info, mostly irrevelant:
Lower bound:
An easy lower bound is Cantor-like set (numbers 1..3^n-1 not containing 1 in their ternary expansion) - its density is n^(log_3 2) (circa 0.631). So any checking if the set isn't too large, and then checking all pairs is not enough to get O(n log n). You have to investigate the sequence smarter. A better lower bound is quoted here - it's n1-c/(log(n))^(1/2). This means Cantor set is not optimal.
Upper bound - my old algorithm:
It is known that for large n, a subset of {1,2,...,n} not containing arithmetic progression has at most n/(log n)^(1/20) elements. The paper On triples in arithmetic progression proves more: the set cannot contain more than n * 228 * (log log n / log n)1/2 elements. So you could check if that bound is achieved and if not, naively check pairs. This is O(n2 * log log n / log n) algorithm, faster than O(n2). Unfortunately "On triples..." is on Springer - but the first page is available, and Ben Green's exposition is available here, page 28, theorem 24.
By the way, the papers are from 1999 - the same year as the first one I mentioned, so that's probably why the first one doesn't mention that result.
This is not a solution, but a similar line of thought to what Olexiy was thinking
I was playing around with creating sequences with maximum number of ones, and they are all quite interesting, I got up to 125 digits and here are the first 3 numbers it found by attempting to insert as many '1' bits as possible:
11011000011011000000000000001101100001101100000000000000000000000000000000000000000110110000110110000000000000011011000011011
10110100010110100000000000010110100010110100000000000000000000000000000000000000000101101000101101000000000000101101000101101
10011001010011001000000000010011001010011001000000000000000000000000000000000000010011001010011001000000000010011001010011001
Notice they are all fractals (not too surprising given the constraints). There may be something in thinking backwards, perhaps if the string is not a fractal of with a characteristic, then it must have a repeating pattern?
Thanks to beta for the better term to describe these numbers.
Update:
Alas it looks like the pattern breaks down when starting with a large enough initial string, such as: 10000000000001:
100000000000011
10000000000001101
100000000000011011
10000000000001101100001
100000000000011011000011
10000000000001101100001101
100000000000011011000011010000000001
100000000000011011000011010000000001001
1000000000000110110000110100000000010011
1000000000000110110000110100000000010011001
10000000000001101100001101000000000100110010000000001
10000000000001101100001101000000000100110010000000001000001
1000000000000110110000110100000000010011001000000000100000100000000000001
10000000000001101100001101000000000100110010000000001000001000000000000011
1000000000000110110000110100000000010011001000000000100000100000000000001101
100000000000011011000011010000000001001100100000000010000010000000000000110100001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001
1000000000000110110000110100000000010011001000000000100000100000000000001101000010010000010000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001
1000000000000110110000110100000000010011001000000000100000100000000000001101000010010000010000001100010000000010000000000000000000000000000000000000000100000010000000000000011
1000000000000110110000110100000000010011001000000000100000100000000000001101000010010000010000001100010000000010000000000000000000000000000000000000000100000010000000000000011000000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001001000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001001000100100000100000000000001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001001000001000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001001000100100000100000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001001000001000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000011
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001001000001000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000011000001
1000000000000110110000110100000000010011001000000000100000100000000000001101000010010000010000001100010000000010000000000000000000000000000000000000000100000010000000000000011000000001100100000000100100000000000010000000010000100000100100010010000010000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000110000010000000000000000000001
1000000000000110110000110100000000010011001000000000100000100000000000001101000010010000010000001100010000000010000000000000000000000000000000000000000100000010000000000000011000000001100100000000100100000000000010000000010000100000100100010010000010000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000110000010000000000000000000001001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001001000100100000100000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000001100000100000000000000000000010010000000000000000000000000000000000001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001001000001000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000011000001000000000000000000000100100000000000000000000000000000000000011
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001001000001000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000011000001000000000000000000000100100000000000000000000000000000000000011001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001001000100100000100000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000001100000100000000000000000000010010000000000000000000000000000000000001100100000000000000000000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001001000100100000100000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000001100000100000000000000000000010010000000000000000000000000000000000001100100000000000000000000001001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001001000100100000100000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000001100000100000000000000000000010010000000000000000000000000000000000001100100000000000000000000001001000001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001001000001000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000011000001000000000000000000000100100000000000000000000000000000000000011001000000000000000000000010010000010000001
1000000000000110110000110100000000010011001000000000100000100000000000001101000010010000010000001100010000000010000000000000000000000000000000000000000100000010000000000000011000000001100100000000100100000000000010000000010000100000100100010010000010000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000110000010000000000000000000001001000000000000000000000000000000000000110010000000000000000000000100100000100000011
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001001000100100000100000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000001100000100000000000000000000010010000000000000000000000000000000000001100100000000000000000000001001000001000000110000000000001
I suspect that a simple approach that looks like O(n^2) will actually yield something better, like O(n ln(n)). The sequences that take the longest to test (for any given n) are the ones that contain no trios, and that puts severe restrictions on the number of 1's that can be in the sequence.
I've come up with some hand-waving arguments, but I haven't been able to find a tidy proof. I'm going to take a stab in the dark: the answer is a very clever idea that the professor has known for so long that it's come to seem obvious, but it's much too hard for the students. (Either that or you slept through the lecture that covered it.)
Revision: 2009-10-17 23:00
I've run this on large numbers (like, strings of 20 million) and I now believe this algorithm is not O(n logn). Notwithstanding that, it's a cool enough implementation and contains a number of optimizations that makes it run really fast. It evaluates all the arrangements of binary strings 24 or fewer digits in under 25 seconds.
I've updated the code to include the 0 <= L < M < U <= X-1 observation from earlier today.
Original
This is, in concept, similar to another question I answered. That code also looked at three values in a series and determined if a triplet satisfied a condition. Here is C# code adapted from that:
using System;
using System.Collections.Generic;
namespace StackOverflow1560523
{
class Program
{
public struct Pair<T>
{
public T Low, High;
}
static bool FindCandidate(int candidate,
List<int> arr,
List<int> pool,
Pair<int> pair,
ref int iterations)
{
int lower = pair.Low, upper = pair.High;
while ((lower >= 0) && (upper < pool.Count))
{
int lowRange = candidate - arr[pool[lower]];
int highRange = arr[pool[upper]] - candidate;
iterations++;
if (lowRange < highRange)
lower -= 1;
else if (lowRange > highRange)
upper += 1;
else
return true;
}
return false;
}
static List<int> BuildOnesArray(string s)
{
List<int> arr = new List<int>();
for (int i = 0; i < s.Length; i++)
if (s[i] == '1')
arr.Add(i);
return arr;
}
static void BuildIndexes(List<int> arr,
ref List<int> even, ref List<int> odd,
ref List<Pair<int>> evenIndex, ref List<Pair<int>> oddIndex)
{
for (int i = 0; i < arr.Count; i++)
{
bool isEven = (arr[i] & 1) == 0;
if (isEven)
{
evenIndex.Add(new Pair<int> {Low=even.Count-1, High=even.Count+1});
oddIndex.Add(new Pair<int> {Low=odd.Count-1, High=odd.Count});
even.Add(i);
}
else
{
oddIndex.Add(new Pair<int> {Low=odd.Count-1, High=odd.Count+1});
evenIndex.Add(new Pair<int> {Low=even.Count-1, High=even.Count});
odd.Add(i);
}
}
}
static int FindSpacedOnes(string s)
{
// List of indexes of 1s in the string
List<int> arr = BuildOnesArray(s);
//if (s.Length < 3)
// return 0;
// List of indexes to odd indexes in arr
List<int> odd = new List<int>(), even = new List<int>();
// evenIndex has indexes into arr to bracket even numbers
// oddIndex has indexes into arr to bracket odd numbers
List<Pair<int>> evenIndex = new List<Pair<int>>(),
oddIndex = new List<Pair<int>>();
BuildIndexes(arr,
ref even, ref odd,
ref evenIndex, ref oddIndex);
int iterations = 0;
for (int i = 1; i < arr.Count-1; i++)
{
int target = arr[i];
bool found = FindCandidate(target, arr, odd, oddIndex[i], ref iterations) ||
FindCandidate(target, arr, even, evenIndex[i], ref iterations);
if (found)
return iterations;
}
return iterations;
}
static IEnumerable<string> PowerSet(int n)
{
for (long i = (1L << (n-1)); i < (1L << n); i++)
{
yield return Convert.ToString(i, 2).PadLeft(n, '0');
}
}
static void Main(string[] args)
{
for (int i = 5; i < 64; i++)
{
int c = 0;
string hardest_string = "";
foreach (string s in PowerSet(i))
{
int cost = find_spaced_ones(s);
if (cost > c)
{
hardest_string = s;
c = cost;
Console.Write("{0} {1} {2}\r", i, c, hardest_string);
}
}
Console.WriteLine("{0} {1} {2}", i, c, hardest_string);
}
}
}
}
The principal differences are:
Exhaustive search of solutions
This code generates a power set of data to find the hardest input to solve for this algorithm.
All solutions versus hardest to solve
The code for the previous question generated all the solutions using a python generator. This code just displays the hardest for each pattern length.
Scoring algorithm
This code checks the distance from the middle element to its left- and right-hand edge. The python code tested whether a sum was above or below 0.
Convergence on a candidate
The current code works from the middle towards the edge to find a candidate. The code in the previous problem worked from the edges towards the middle. This last change gives a large performance improvement.
Use of even and odd pools
Based on the observations at the end of this write-up, the code searches pairs of even numbers of pairs of odd numbers to find L and U, keeping M fixed. This reduces the number of searches by pre-computing information. Accordingly, the code uses two levels of indirection in the main loop of FindCandidate and requires two calls to FindCandidate for each middle element: once for even numbers and once for odd ones.
The general idea is to work on indexes, not the raw representation of the data. Calculating an array where the 1's appear allows the algorithm to run in time proportional to the number of 1's in the data rather than in time proportional to the length of the data. This is a standard transformation: create a data structure that allows faster operation while keeping the problem equivalent.
The results are out of date: removed.
Edit: 2009-10-16 18:48
On yx's data, which is given some credence in the other responses as representative of hard data to calculate on, I get these results... I removed these. They are out of date.
I would point out that this data is not the hardest for my algorithm, so I think the assumption that yx's fractals are the hardest to solve is mistaken. The worst case for a particular algorithm, I expect, will depend upon the algorithm itself and will not likely be consistent across different algorithms.
Edit: 2009-10-17 13:30
Further observations on this.
First, convert the string of 0's and 1's into an array of indexes for each position of the 1's. Say the length of that array A is X. Then the goal is to find
0 <= L < M < U <= X-1
such that
A[M] - A[L] = A[U] - A[M]
or
2*A[M] = A[L] + A[U]
Since A[L] and A[U] sum to an even number, they can't be (even, odd) or (odd, even). The search for a match could be improved by splitting A[] into odd and even pools and searching for matches on A[M] in the pools of odd and even candidates in turn.
However, this is more of a performance optimization than an algorithmic improvement, I think. The number of comparisons should drop, but the order of the algorithm should be the same.
Edit 2009-10-18 00:45
Yet another optimization occurs to me, in the same vein as separating the candidates into even and odd. Since the three indexes have to add to a multiple of 3 (a, a+x, a+2x -- mod 3 is 0, regardless of a and x), you can separate L, M, and U into their mod 3 values:
M L U
0 0 0
1 2
2 1
1 0 2
1 1
2 0
2 0 1
1 0
2 2
In fact, you could combine this with the even/odd observation and separate them into their mod 6 values:
M L U
0 0 0
1 5
2 4
3 3
4 2
5 1
and so on. This would provide a further performance optimization but not an algorithmic speedup.
Wasn't able to come up with the solution yet :(, but have some ideas.
What if we start from a reverse problem: construct a sequence with the maximum number of 1s and WITHOUT any evenly spaced trios. If you can prove the maximum number of 1s is o(n), then you can improve your estimate by iterating only through list of 1s only.
This may help....
This problem reduces to the following:
Given a sequence of positive integers, find a contiguous subsequence partitioned into a prefix and a suffix such that the sum of the prefix of the subsequence is equal to the sum of the suffix of the subsequence.
For example, given a sequence of [ 3, 5, 1, 3, 6, 5, 2, 2, 3, 5, 6, 4 ], we would find a subsequence of [ 3, 6, 5, 2, 2] with a prefix of [ 3, 6 ] with prefix sum of 9 and a suffix of [ 5, 2, 2 ] with suffix sum of 9.
The reduction is as follows:
Given a sequence of zeros and ones, and starting at the leftmost one, continue moving to the right. Each time another one is encountered, record the number of moves since the previous one was encountered and append that number to the resulting sequence.
For example, given a sequence of [ 0, 1, 1, 0, 0, 1, 0, 0, 0, 1 0 ], we would find the reduction of [ 1, 3, 4]. From this reduction, we calculate the contiguous subsequence of [ 1, 3, 4], the prefix of [ 1, 3] with sum of 4, and the suffix of [ 4 ] with sum of 4.
This reduction may be computed in O(n).
Unfortunately, I am not sure where to go from here.
For the simple problem type (i.e. you search three "1" with only (i.e. zero or more) "0" between it), Its quite simple: You could just split the sequence at every "1" and look for two adjacent subsequences having the same length (the second subsequence not being the last one, of course). Obviously, this can be done in O(n) time.
For the more complex version (i.e. you search an index i and an gap g>0 such that s[i]==s[i+g]==s[i+2*g]=="1"), I'm not sure, if there exists an O(n log n) solution, since there are possibly O(n²) triplets having this property (think of a string of all ones, there are approximately n²/2 such triplets). Of course, you are looking for only one of these, but I have currently no idea, how to find it ...
A fun question, but once you realise that the actual pattern between two '1's does not matter, the algorithm becomes:
scan look for a '1'
starting from the next position scan for another '1' (to the end of the array minus the distance from the current first '1' or else the 3rd '1' would be out of bounds)
if at the position of the 2nd '1' plus the distance to the first 1' a third '1' is found, we have evenly spaces ones.
In code, JTest fashion, (Note this code isn't written to be most efficient and I added some println's to see what happens.)
import java.util.Random;
import junit.framework.TestCase;
public class AlgorithmTest extends TestCase {
/**
* Constructor for GetNumberTest.
*
* #param name The test's name.
*/
public AlgorithmTest(String name) {
super(name);
}
/**
* #see TestCase#setUp()
*/
protected void setUp() throws Exception {
super.setUp();
}
/**
* #see TestCase#tearDown()
*/
protected void tearDown() throws Exception {
super.tearDown();
}
/**
* Tests the algorithm.
*/
public void testEvenlySpacedOnes() {
assertFalse(isEvenlySpaced(1));
assertFalse(isEvenlySpaced(0x058003));
assertTrue(isEvenlySpaced(0x07001));
assertTrue(isEvenlySpaced(0x01007));
assertTrue(isEvenlySpaced(0x101010));
// some fun tests
Random random = new Random();
isEvenlySpaced(random.nextLong());
isEvenlySpaced(random.nextLong());
isEvenlySpaced(random.nextLong());
}
/**
* #param testBits
*/
private boolean isEvenlySpaced(long testBits) {
String testString = Long.toBinaryString(testBits);
char[] ones = testString.toCharArray();
final char ONE = '1';
for (int n = 0; n < ones.length - 1; n++) {
if (ONE == ones[n]) {
for (int m = n + 1; m < ones.length - m + n; m++) {
if (ONE == ones[m] && ONE == ones[m + m - n]) {
System.out.println(" IS evenly spaced: " + testBits + '=' + testString);
System.out.println(" at: " + n + ", " + m + ", " + (m + m - n));
return true;
}
}
}
}
System.out.println("NOT evenly spaced: " + testBits + '=' + testString);
return false;
}
}
I thought of a divide-and-conquer approach that might work.
First, in preprocessing you need to insert all numbers less than one half your input size (n/3) into a list.
Given a string: 0000010101000100 (note that this particular example is valid)
Insert all primes (and 1) from 1 to (16/2) into a list: {1, 2, 3, 4, 5, 6, 7}
Then divide it in half:
100000101 01000100
Keep doing this until you get to strings of size 1. For all size-one strings with a 1 in them, add the index of the string to the list of possibilities; otherwise, return -1 for failure.
You'll also need to return a list of still-possible spacing distances, associated with each starting index. (Start with the list you made above and remove numbers as you go) Here, an empty list means you're only dealing with one 1 and so any spacing is possible at this point; otherwise the list includes spacings that must be ruled out.
So continuing with the example above:
1000 0101 0100 0100
10 00 01 01 01 00 01 00
1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0
In the first combine step, we have eight sets of two now. In the first, we have the possibility of a set, but we learn that spacing by 1 is impossible because of the other zero being there. So we return 0 (for the index) and {2,3,4,5,7} for the fact that spacing by 1 is impossible. In the second, we have nothing and so return -1. In the third we have a match with no spacings eliminated in index 5, so return 5, {1,2,3,4,5,7}. In the fourth pair we return 7, {1,2,3,4,5,7}. In the fifth, return 9, {1,2,3,4,5,7}. In the sixth, return -1. In the seventh, return 13, {1,2,3,4,5,7}. In the eighth, return -1.
Combining again into four sets of four, we have:
1000: Return (0, {4,5,6,7})
0101: Return (5, {2,3,4,5,6,7}), (7, {1,2,3,4,5,6,7})
0100: Return (9, {3,4,5,6,7})
0100: Return (13, {3,4,5,6,7})
Combining into sets of eight:
10000101: Return (0, {5,7}), (5, {2,3,4,5,6,7}), (7, {1,2,3,4,5,6,7})
01000100: Return (9, {4,7}), (13, {3,4,5,6,7})
Combining into a set of sixteen:
10000101 01000100
As we've progressed, we keep checking all the possibilities so far. Up to this step we've left stuff that went beyond the end of the string, but now we can check all the possibilities.
Basically, we check the first 1 with spacings of 5 and 7, and find that they don't line up to 1's. (Note that each check is CONSTANT, not linear time) Then we check the second one (index 5) with spacings of 2, 3, 4, 5, 6, and 7-- or we would, but we can stop at 2 since that actually matches up.
Phew! That's a rather long algorithm.
I don't know 100% if it's O(n log n) because of the last step, but everything up to there is definitely O(n log n) as far as I can tell. I'll get back to this later and try to refine the last step.
EDIT: Changed my answer to reflect Welbog's comment. Sorry for the error. I'll write some pseudocode later, too, when I get a little more time to decipher what I wrote again. ;-)
I'll give my rough guess here, and let those who are better with calculating complexity to help me on how my algorithm fares in O-notation wise
given binary string 0000010101000100 (as example)
crop head and tail of zeroes -> 00000 101010001 00
we get 101010001 from previous calculation
check if the middle bit is 'one', if true, found valid three evenly spaced 'ones' (only if the number of bits is odd numbered)
correlatively, if the remained cropped number of bits is even numbered, the head and tail 'one' cannot be part of evenly spaced 'one',
we use 1010100001 as example (with an extra 'zero' to become even numbered crop), in this case we need to crop again, then becomes -> 10101 00001
we get 10101 from previous calculation, and check middle bit, and we found the evenly spaced bit again
I have no idea how to calculate complexity for this, can anyone help?
edit: add some code to illustrate my idea
edit2: tried to compile my code and found some major mistakes, fixed
char *binaryStr = "0000010101000100";
int main() {
int head, tail, pos;
head = 0;
tail = strlen(binaryStr)-1;
if( (pos = find3even(head, tail)) >=0 )
printf("found it at position %d\n", pos);
return 0;
}
int find3even(int head, int tail) {
int pos = 0;
if(head >= tail) return -1;
while(binaryStr[head] == '0')
if(head<tail) head++;
while(binaryStr[tail] == '0')
if(head<tail) tail--;
if(head >= tail) return -1;
if( (tail-head)%2 == 0 && //true if odd numbered
(binaryStr[head + (tail-head)/2] == '1') ) {
return head;
}else {
if( (pos = find3even(head, tail-1)) >=0 )
return pos;
if( (pos = find3even(head+1, tail)) >=0 )
return pos;
}
return -1;
}
I came up with something like this:
def IsSymetric(number):
number = number.strip('0')
if len(number) < 3:
return False
if len(number) % 2 == 0:
return IsSymetric(number[1:]) or IsSymetric(number[0:len(number)-2])
else:
if number[len(number)//2] == '1':
return True
return IsSymetric(number[:(len(number)//2)]) or IsSymetric(number[len(number)//2+1:])
return False
This is inspired by andycjw.
Truncate the zeros.
If even then test two substring 0 - (len-2) (skip last character) and from 1 - (len-1) (skip the first char)
If not even than if the middle char is one than we have success. Else divide the string in the midle without the midle element and check both parts.
As to the complexity this might be O(nlogn) as in each recursion we are dividing by two.
Hope it helps.
Ok, I'm going to take another stab at the problem. I think I can prove a O(n log(n)) algorithm that is similar to those already discussed by using a balanced binary tree to store distances between 1's. This approach was inspired by Justice's observation about reducing the problem to a list of distances between the 1's.
Could we scan the input string to construct a balanced binary tree around the position of 1's such that each node stores the position of the 1 and each edge is labeled with the distance to the adjacent 1 for each child node. For example:
10010001 gives the following tree
3
/ \
2 / \ 3
/ \
0 7
This can be done in O(n log(n)) since, for a string of size n, each insertion takes O(log(n)) in the worst case.
Then the problem is to search the tree to discover whether, at any node, there is a path from that node through the left-child that has the same distance as a path through the right child. This can be done recursively on each subtree. When merging two subtrees in the search, we must compare the distances from paths in the left subtree with distances from paths in the right. Since the number of paths in a subtree will be proportional to log(n), and the number of nodes is n, I believe this can be done in O(n log(n)) time.
Did I miss anything?
This seemed liked a fun problem so I decided to try my hand at it.
I am making the assumption that 111000001 would find the first 3 ones and be successful. Essentially the number of zeroes following the 1 is the important thing, since 0111000 is the same as 111000 according to your definition. Once you find two cases of 1, the next 1 found completes the trilogy.
Here it is in Python:
def find_three(bstring):
print bstring
dict = {}
lastone = -1
zerocount = 0
for i in range(len(bstring)):
if bstring[i] == '1':
print i, ': 1'
if lastone != -1:
if(zerocount in dict):
dict[zerocount].append(lastone)
if len(dict[zerocount]) == 2:
dict[zerocount].append(i)
return True, dict
else:
dict[zerocount] = [lastone]
lastone = i
zerocount = 0
else:
zerocount = zerocount + 1
#this is really just book keeping, as we have failed at this point
if lastone != -1:
if(zerocount in dict):
dict[zerocount].append(lastone)
else:
dict[zerocount] = [lastone]
return False, dict
This is a first try, so I'm sure this could be written in a cleaner manner. Please list the cases where this method fails down below.
I assume the reason this is nlog(n) is due to the following:
To find the 1 that is the start of the triplet, you need to check (n-2) characters. If you haven't found it by that point, you won't (chars n-1 and n cannot start a triplet) (O(n))
To find the second 1 that is the part of the triplet (started by the first one), you need to check m/2 (m=n-x, where x is the offset of the first 1) characters. This is because, if you haven't found the second 1 by the time you're halfway from the first one to the end, you won't... since the third 1 must be exactly the same distance past the second. (O(log(n)))
It O(1) to find the last 1 since you know the index it must be at by the time you find the first and second.
So, you have n, log(n), and 1... O(nlogn)
Edit: Oops, my bad. My brain had it set that n/2 was logn... which it obviously isn't (doubling the number on items still doubles the number of iterations on the inner loop). This is still at n^2, not solving the problem. Well, at least I got to write some code :)
Implementation in Tcl
proc get-triplet {input} {
for {set first 0} {$first < [string length $input]-2} {incr first} {
if {[string index $input $first] != 1} {
continue
}
set start [expr {$first + 1}]
set end [expr {1+ $first + (([string length $input] - $first) /2)}]
for {set second $start} {$second < $end} {incr second} {
if {[string index $input $second] != 1} {
continue
}
set last [expr {($second - $first) + $second}]
if {[string index $input $last] == 1} {
return [list $first $second $last]
}
}
}
return {}
}
get-triplet 10101 ;# 0 2 4
get-triplet 10111 ;# 0 2 4
get-triplet 11100000 ;# 0 1 2
get-triplet 0100100100 ;# 1 4 7
I think I have found a way of solving the problem, but I can't construct a formal proof. The solution I made is written in Java, and it uses a counter 'n' to count how many list/array accesses it does. So n should be less than or equal to stringLength*log(stringLength) if it is correct. I tried it for the numbers 0 to 2^22, and it works.
It starts by iterating over the input string and making a list of all the indexes which hold a one. This is just O(n).
Then from the list of indexes it picks a firstIndex, and a secondIndex which is greater than the first. These two indexes must hold ones, because they are in the list of indexes. From there the thirdIndex can be calculated. If the inputString[thirdIndex] is a 1 then it halts.
public static int testString(String input){
//n is the number of array/list accesses in the algorithm
int n=0;
//Put the indices of all the ones into a list, O(n)
ArrayList<Integer> ones = new ArrayList<Integer>();
for(int i=0;i<input.length();i++){
if(input.charAt(i)=='1'){
ones.add(i);
}
}
//If less than three ones in list, just stop
if(ones.size()<3){
return n;
}
int firstIndex, secondIndex, thirdIndex;
for(int x=0;x<ones.size()-2;x++){
n++;
firstIndex = ones.get(x);
for(int y=x+1; y<ones.size()-1; y++){
n++;
secondIndex = ones.get(y);
thirdIndex = secondIndex*2 - firstIndex;
if(thirdIndex >= input.length()){
break;
}
n++;
if(input.charAt(thirdIndex) == '1'){
//This case is satisfied if it has found three evenly spaced ones
//System.out.println("This one => " + input);
return n;
}
}
}
return n;
}
additional note: the counter n is not incremented when it iterates over the input string to construct the list of indexes. This operation is O(n), so it won't have an effect on the algorithm complexity anyway.
One inroad into the problem is to think of factors and shifting.
With shifting, you compare the string of ones and zeroes with a shifted version of itself. You then take matching ones. Take this example shifted by two:
1010101010
1010101010
------------
001010101000
The resulting 1's (bitwise ANDed), must represent all those 1's which are evenly spaced by two. The same example shifted by three:
1010101010
1010101010
-------------
0000000000000
In this case there are no 1's which are evenly spaced three apart.
So what does this tell you? Well that you only need to test shifts which are prime numbers. For example say you have two 1's which are six apart. You would only have to test 'two' shifts and 'three' shifts (since these divide six). For example:
10000010
10000010 (Shift by two)
10000010
10000010 (We have a match)
10000010
10000010 (Shift by three)
10000010 (We have a match)
So the only shifts you ever need to check are 2,3,5,7,11,13 etc. Up to the prime closest to the square root of size of the string of digits.
Nearly solved?
I think I am closer to a solution. Basically:
Scan the string for 1's. For each 1 note it's remainder after taking a modulus of its position. The modulus ranges from 1 to half the size of the string. This is because the largest possible separation size is half the string. This is done in O(n^2). BUT. Only prime moduli need be checked so O(n^2/log(n))
Sort the list of modulus/remainders in order largest modulus first, this can be done in O(n*log(n)) time.
Look for three consecutive moduli/remainders which are the same.
Somehow retrieve the position of the ones!
I think the biggest clue to the answer, is that the fastest sort algorithms, are O(n*log(n)).
WRONG
Step 1 is wrong as pointed out by a colleague. If we have 1's at position 2,12 and 102. Then taking a modulus of 10, they would all have the same remainders, and yet are not equally spaced apart! Sorry.
Here are some thoughts that, despite my best efforts, will not seem to wrap themselves up in a bow. Still, they might be a useful starting point for someone's analysis.
Consider the proposed solution as follows, which is the approach that several folks have suggested, including myself in a prior version of this answer. :)
Trim leading and trailing zeroes.
Scan the string looking for 1's.
When a 1 is found:
Assume that it is the middle 1 of the solution.
For each prior 1, use its saved position to compute the anticipated position of the final 1.
If the computed position is after the end of the string it cannot be part of the solution, so drop the position from the list of candidates.
Check the solution.
If the solution was not found, add the current 1 to the list of candidates.
Repeat until no more 1's are found.
Now consider input strings strings like the following, which will not have a solution:
101
101001
1010010001
101001000100001
101001000100001000001
In general, this is the concatenation of k strings of the form j 0's followed by a 1 for j from zero to k-1.
k=2 101
k=3 101001
k=4 1010010001
k=5 101001000100001
k=6 101001000100001000001
Note that the lengths of the substrings are 1, 2, 3, etc. So, problem size n has substrings of lengths 1 to k such that n = k(k+1)/2.
k=2 n= 3 101
k=3 n= 6 101001
k=4 n=10 1010010001
k=5 n=15 101001000100001
k=6 n=21 101001000100001000001
Note that k also tracks the number of 1's that we have to consider. Remember that every time we see a 1, we need to consider all the 1's seen so far. So when we see the second 1, we only consider the first, when we see the third 1, we reconsider the first two, when we see the fourth 1, we need to reconsider the first three, and so on. By the end of the algorithm, we've considered k(k-1)/2 pairs of 1's. Call that p.
k=2 n= 3 p= 1 101
k=3 n= 6 p= 3 101001
k=4 n=10 p= 6 1010010001
k=5 n=15 p=10 101001000100001
k=6 n=21 p=15 101001000100001000001
The relationship between n and p is that n = p + k.
The process of going through the string takes O(n) time. Each time a 1 is encountered, a maximum of (k-1) comparisons are done. Since n = k(k+1)/2, n > k**2, so sqrt(n) > k. This gives us O(n sqrt(n)) or O(n**3/2). Note however that may not be a really tight bound, because the number of comparisons goes from 1 to a maximum of k, it isn't k the whole time. But I'm not sure how to account for that in the math.
It still isn't O(n log(n)). Also, I can't prove those inputs are the worst cases, although I suspect they are. I think a denser packing of 1's to the front results in an even sparser packing at the end.
Since someone may still find it useful, here's my code for that solution in Perl:
#!/usr/bin/perl
# read input as first argument
my $s = $ARGV[0];
# validate the input
$s =~ /^[01]+$/ or die "invalid input string\n";
# strip leading and trailing 0's
$s =~ s/^0+//;
$s =~ s/0+$//;
# prime the position list with the first '1' at position 0
my #p = (0);
# start at position 1, which is the second character
my $i = 1;
print "the string is $s\n\n";
while ($i < length($s)) {
if (substr($s, $i, 1) eq '1') {
print "found '1' at position $i\n";
my #t = ();
# assuming this is the middle '1', go through the positions
# of all the prior '1's and check whether there's another '1'
# in the correct position after this '1' to make a solution
while (scalar #p) {
# $p is the position of the prior '1'
my $p = shift #p;
# $j is the corresponding position for the following '1'
my $j = 2 * $i - $p;
# if $j is off the end of the string then we don't need to
# check $p anymore
next if ($j >= length($s));
print "checking positions $p, $i, $j\n";
if (substr($s, $j, 1) eq '1') {
print "\nsolution found at positions $p, $i, $j\n";
exit 0;
}
# if $j isn't off the end of the string, keep $p for next time
push #t, $p;
}
#p = #t;
# add this '1' to the list of '1' positions
push #p, $i;
}
$i++;
}
print "\nno solution found\n";
While scanning 1s, add their positions to a List. When adding the second and successive 1s, compare them to each position in the list so far. Spacing equals currentOne (center) - previousOne (left). The right-side bit is currentOne + spacing. If it's 1, the end.
The list of ones grows inversely with the space between them. Simply stated, if you've got a lot of 0s between the 1s (as in a worst case), your list of known 1s will grow quite slowly.
using System;
using System.Collections.Generic;
namespace spacedOnes
{
class Program
{
static int[] _bits = new int[8] {128, 64, 32, 16, 8, 4, 2, 1};
static void Main(string[] args)
{
var bytes = new byte[4];
var r = new Random();
r.NextBytes(bytes);
foreach (var b in bytes) {
Console.Write(getByteString(b));
}
Console.WriteLine();
var bitCount = bytes.Length * 8;
var done = false;
var onePositions = new List<int>();
for (var i = 0; i < bitCount; i++)
{
if (isOne(bytes, i)) {
if (onePositions.Count > 0) {
foreach (var knownOne in onePositions) {
var spacing = i - knownOne;
var k = i + spacing;
if (k < bitCount && isOne(bytes, k)) {
Console.WriteLine("^".PadLeft(knownOne + 1) + "^".PadLeft(spacing) + "^".PadLeft(spacing));
done = true;
break;
}
}
}
if (done) {
break;
}
onePositions.Add(i);
}
}
Console.ReadKey();
}
static String getByteString(byte b) {
var s = new char[8];
for (var i=0; i<s.Length; i++) {
s[i] = ((b & _bits[i]) > 0 ? '1' : '0');
}
return new String(s);
}
static bool isOne(byte[] bytes, int i)
{
var byteIndex = i / 8;
var bitIndex = i % 8;
return (bytes[byteIndex] & _bits[bitIndex]) > 0;
}
}
}
I thought I'd add one comment before posting the 22nd naive solution to the problem. For the naive solution, we don't need to show that the number of 1's in the string is at most O(log(n)), but rather that it is at most O(sqrt(n*log(n)).
Solver:
def solve(Str):
indexes=[]
#O(n) setup
for i in range(len(Str)):
if Str[i]=='1':
indexes.append(i)
#O((number of 1's)^2) processing
for i in range(len(indexes)):
for j in range(i+1, len(indexes)):
indexDiff = indexes[j] - indexes[i]
k=indexes[j] + indexDiff
if k<len(Str) and Str[k]=='1':
return True
return False
It's basically a fair bit similar to flybywire's idea and implementation, though looking ahead instead of back.
Greedy String Builder:
#assumes final char hasn't been added, and would be a 1
def lastCharMakesSolvable(Str):
endIndex=len(Str)
j=endIndex-1
while j-(endIndex-j) >= 0:
k=j-(endIndex-j)
if k >= 0 and Str[k]=='1' and Str[j]=='1':
return True
j=j-1
return False
def expandString(StartString=''):
if lastCharMakesSolvable(StartString):
return StartString + '0'
return StartString + '1'
n=1
BaseStr=""
lastCount=0
while n<1000000:
BaseStr=expandString(BaseStr)
count=BaseStr.count('1')
if count != lastCount:
print(len(BaseStr), count)
lastCount=count
n=n+1
(In my defense, I'm still in the 'learn python' stage of understanding)
Also, potentially useful output from the greedy building of strings, there's a rather consistent jump after hitting a power of 2 in the number of 1's... which I was not willing to wait around to witness hitting 2096.
strlength # of 1's
1 1
2 2
4 3
5 4
10 5
14 8
28 9
41 16
82 17
122 32
244 33
365 64
730 65
1094 128
2188 129
3281 256
6562 257
9842 512
19684 513
29525 1024
I'll try to present a mathematical approach. This is more a beginning than an end, so any help, comment, or even contradiction - will be deeply appreciated. However, if this approach is proven - the algorithm is a straight-forward search in the string.
Given a fixed number of spaces k and a string S, the search for a k-spaced-triplet takes O(n) - We simply test for every 0<=i<=(n-2k) if S[i]==S[i+k]==S[i+2k]. The test takes O(1) and we do it n-k times where k is a constant, so it takes O(n-k)=O(n).
Let us assume that there is an Inverse Proportion between the number of 1's and the maximum spaces we need to search for. That is, If there are many 1's, there must be a triplet and it must be quite dense; If there are only few 1's, The triplet (if any) can be quite sparse. In other words, I can prove that if I have enough 1's, such triplet must exist - and the more 1's I have, a more dense triplet must be found. This can be explained by the Pigeonhole principle - Hope to elaborate on this later.
Say have an upper bound k on the possible number of spaces I have to look for. Now, for each 1 located in S[i] we need to check for 1 in S[i-1] and S[i+1], S[i-2] and S[i+2], ... S[i-k] and S[i+k]. This takes O((k^2-k)/2)=O(k^2) for each 1 in S - due to Gauss' Series Summation Formula. Note that this differs from section 1 - I'm having k as an upper bound for the number of spaces, not as a constant space.
We need to prove O(n*log(n)). That is, we need to show that k*(number of 1's) is proportional to log(n).
If we can do that, the algorithm is trivial - for each 1 in S whose index is i, simply look for 1's from each side up to distance k. If two were found in the same distance, return i and k. Again, the tricky part would be finding k and proving the correctness.
I would really appreciate your comments here - I have been trying to find the relation between k and the number of 1's on my whiteboard, so far without success.
Assumption:
Just wrong, talking about log(n) number of upper limit of ones
EDIT:
Now I found that using Cantor numbers (if correct), density on set is (2/3)^Log_3(n) (what a weird function) and I agree, log(n)/n density is to strong.
If this is upper limit, there is algorhitm who solves this problem in at least O(n*(3/2)^(log(n)/log(3))) time complexity and O((3/2)^(log(n)/log(3))) space complexity. (check Justice's answer for algorhitm)
This is still by far better than O(n^2)
This function ((3/2)^(log(n)/log(3))) really looks like n*log(n) on first sight.
How did I get this formula?
Applaying Cantors number on string.
Supose that length of string is 3^p == n
At each step in generation of Cantor string you keep 2/3 of prevous number of ones. Apply this p times.
That mean (n * ((2/3)^p)) -> (((3^p)) * ((2/3)^p)) remaining ones and after simplification 2^p.
This mean 2^p ones in 3^p string -> (3/2)^p ones . Substitute p=log(n)/log(3) and get
((3/2)^(log(n)/log(3)))
How about a simple O(n) solution, with O(n^2) space? (Uses the assumption that all bitwise operators work in O(1).)
The algorithm basically works in four stages:
Stage 1: For each bit in your original number, find out how far away the ones are, but consider only one direction. (I considered all the bits in the direction of the least significant bit.)
Stage 2: Reverse the order of the bits in the input;
Stage 3: Re-run step 1 on the reversed input.
Stage 4: Compare the results from Stage 1 and Stage 3. If any bits are equally spaced above AND below we must have a hit.
Keep in mind that no step in the above algorithm takes longer than O(n). ^_^
As an added benefit, this algorithm will find ALL equally spaced ones from EVERY number. So for example if you get a result of "0x0005" then there are equally spaced ones at BOTH 1 and 3 units away
I didn't really try optimizing the code below, but it is compilable C# code that seems to work.
using System;
namespace ThreeNumbers
{
class Program
{
const int uint32Length = 32;
static void Main(string[] args)
{
Console.Write("Please enter your integer: ");
uint input = UInt32.Parse(Console.ReadLine());
uint[] distancesLower = Distances(input);
uint[] distancesHigher = Distances(Reverse(input));
PrintHits(input, distancesLower, distancesHigher);
}
/// <summary>
/// Returns an array showing how far the ones away from each bit in the input. Only
/// considers ones at lower signifcant bits. Index 0 represents the least significant bit
/// in the input. Index 1 represents the second least significant bit in the input and so
/// on. If a one is 3 away from the bit in question, then the third least significant bit
/// of the value will be sit.
///
/// As programed this algorithm needs: O(n) time, and O(n*log(n)) space.
/// (Where n is the number of bits in the input.)
/// </summary>
public static uint[] Distances(uint input)
{
uint[] distanceToOnes = new uint[uint32Length];
uint result = 0;
//Sets how far each bit is from other ones. Going in the direction of LSB to MSB
for (uint bitIndex = 1, arrayIndex = 0; bitIndex != 0; bitIndex <<= 1, ++arrayIndex)
{
distanceToOnes[arrayIndex] = result;
result <<= 1;
if ((input & bitIndex) != 0)
{
result |= 1;
}
}
return distanceToOnes;
}
/// <summary>
/// Reverses the bits in the input.
///
/// As programmed this algorithm needs O(n) time and O(n) space.
/// (Where n is the number of bits in the input.)
/// </summary>
/// <param name="input"></param>
/// <returns></returns>
public static uint Reverse(uint input)
{
uint reversedInput = 0;
for (uint bitIndex = 1; bitIndex != 0; bitIndex <<= 1)
{
reversedInput <<= 1;
reversedInput |= (uint)((input & bitIndex) != 0 ? 1 : 0);
}
return reversedInput;
}
/// <summary>
/// Goes through each bit in the input, to check if there are any bits equally far away in
/// the distancesLower and distancesHigher
/// </summary>
public static void PrintHits(uint input, uint[] distancesLower, uint[] distancesHigher)
{
const int offset = uint32Length - 1;
for (uint bitIndex = 1, arrayIndex = 0; bitIndex != 0; bitIndex <<= 1, ++arrayIndex)
{
//hits checks if any bits are equally spaced away from our current value
bool isBitSet = (input & bitIndex) != 0;
uint hits = distancesLower[arrayIndex] & distancesHigher[offset - arrayIndex];
if (isBitSet && (hits != 0))
{
Console.WriteLine(String.Format("The {0}-th LSB has hits 0x{1:x4} away", arrayIndex + 1, hits));
}
}
}
}
}
Someone will probably comment that for any sufficiently large number, bitwise operations cannot be done in O(1). You'd be right. However, I'd conjecture that every solution that uses addition, subtraction, multiplication, or division (which cannot be done by shifting) would also have that problem.
Below is a solution. There could be some little mistakes here and there, but the idea is sound.
Edit: It's not n * log(n)
PSEUDO CODE:
foreach character in the string
if the character equals 1 {
if length cache > 0 { //we can skip the first one
foreach location in the cache { //last in first out kind of order
if ((currentlocation + (currentlocation - location)) < length string)
if (string[(currentlocation + (currentlocation - location))] equals 1)
return found evenly spaced string
else
break;
}
}
remember the location of this character in a some sort of cache.
}
return didn't find evenly spaced string
C# code:
public static Boolean FindThreeEvenlySpacedOnes(String str) {
List<int> cache = new List<int>();
for (var x = 0; x < str.Length; x++) {
if (str[x] == '1') {
if (cache.Count > 0) {
for (var i = cache.Count - 1; i > 0; i--) {
if ((x + (x - cache[i])) >= str.Length)
break;
if (str[(x + (x - cache[i]))] == '1')
return true;
}
}
cache.Add(x);
}
}
return false;
}
How it works:
iteration 1:
x
|
101101001
// the location of this 1 is stored in the cache
iteration 2:
x
|
101101001
iteration 3:
a x b
| | |
101101001
//we retrieve location a out of the cache and then based on a
//we calculate b and check if te string contains a 1 on location b
//and of course we store x in the cache because it's a 1
iteration 4:
axb
|||
101101001
a x b
| | |
101101001
iteration 5:
x
|
101101001
iteration 6:
a x b
| | |
101101001
a x b
| | |
101101001
//return found evenly spaced string
Obviously we need to at least check bunches of triplets at the same time, so we need to compress the checks somehow. I have a candidate algorithm, but analyzing the time complexity is beyond my ability*time threshold.
Build a tree where each node has three children and each node contains the total number of 1's at its leaves. Build a linked list over the 1's, as well. Assign each node an allowed cost proportional to the range it covers. As long as the time we spend at each node is within budget, we'll have an O(n lg n) algorithm.
--
Start at the root. If the square of the total number of 1's below it is less than its allowed cost, apply the naive algorithm. Otherwise recurse on its children.
Now we have either returned within budget, or we know that there are no valid triplets entirely contained within one of the children. Therefore we must check the inter-node triplets.
Now things get incredibly messy. We essentially want to recurse on the potential sets of children while limiting the range. As soon as the range is constrained enough that the naive algorithm will run under budget, you do it. Enjoy implementing this, because I guarantee it will be tedious. There's like a dozen cases.
--
The reason I think that algorithm will work is because the sequences without valid triplets appear to go alternate between bunches of 1's and lots of 0's. It effectively splits the nearby search space, and the tree emulates that splitting.
The run time of the algorithm is not obvious, at all. It relies on the non-trivial properties of the sequence. If the 1's are really sparse then the naive algorithm will work under budget. If the 1's are dense, then a match should be found right away. But if the density is 'just right' (eg. near ~n^0.63, which you can achieve by setting all bits at positions with no '2' digit in base 3), I don't know if it will work. You would have to prove that the splitting effect is strong enough.
No theoretical answer here, but I wrote a quick Java program to explore the running-time behavior as a function of k and n, where n is the total bit length and k is the number of 1's. I'm with a few of the answerers who are saying that the "regular" algorithm that checks all the pairs of bit positions and looks for the 3rd bit, even though it would require O(k^2) in the worst case, in reality because the worst-case needs sparse bitstrings, is O(n ln n).
Anyway here's the program, below. It's a Monte-Carlo style program which runs a large number of trials NTRIALS for constant n, and randomly generates bitsets for a range of k-values using Bernoulli processes with ones-density constrained between limits that can be specified, and records the running time of finding or failing to find a triplet of evenly spaced ones, time measured in steps NOT in CPU time. I ran it for n=64, 256, 1024, 4096, 16384* (still running), first a test run with 500000 trials to see which k-values take the longest running time, then another test with 5000000 trials with narrowed ones-density focus to see what those values look like. The longest running times do happen with very sparse density (e.g. for n=4096 the running time peaks are in the k=16-64 range, with a gentle peak for mean runtime at 4212 steps # k=31, max runtime peaked at 5101 steps # k=58). It looks like it would take extremely large values of N for the worst-case O(k^2) step to become larger than the O(n) step where you scan the bitstring to find the 1's position indices.
package com.example.math;
import java.io.PrintStream;
import java.util.BitSet;
import java.util.Random;
public class EvenlySpacedOnesTest {
static public class StatisticalSummary
{
private int n=0;
private double min=Double.POSITIVE_INFINITY;
private double max=Double.NEGATIVE_INFINITY;
private double mean=0;
private double S=0;
public StatisticalSummary() {}
public void add(double x) {
min = Math.min(min, x);
max = Math.max(max, x);
++n;
double newMean = mean + (x-mean)/n;
S += (x-newMean)*(x-mean);
// this algorithm for mean,std dev based on Knuth TAOCP vol 2
mean = newMean;
}
public double getMax() { return (n>0)?max:Double.NaN; }
public double getMin() { return (n>0)?min:Double.NaN; }
public int getCount() { return n; }
public double getMean() { return (n>0)?mean:Double.NaN; }
public double getStdDev() { return (n>0)?Math.sqrt(S/n):Double.NaN; }
// some may quibble and use n-1 for sample std dev vs population std dev
public static void printOut(PrintStream ps, StatisticalSummary[] statistics) {
for (int i = 0; i < statistics.length; ++i)
{
StatisticalSummary summary = statistics[i];
ps.printf("%d\t%d\t%.0f\t%.0f\t%.5f\t%.5f\n",
i,
summary.getCount(),
summary.getMin(),
summary.getMax(),
summary.getMean(),
summary.getStdDev());
}
}
}
public interface RandomBernoulliProcess // see http://en.wikipedia.org/wiki/Bernoulli_process
{
public void setProbability(double d);
public boolean getNextBoolean();
}
static public class Bernoulli implements RandomBernoulliProcess
{
final private Random r = new Random();
private double p = 0.5;
public boolean getNextBoolean() { return r.nextDouble() < p; }
public void setProbability(double d) { p = d; }
}
static public class TestResult {
final public int k;
final public int nsteps;
public TestResult(int k, int nsteps) { this.k=k; this.nsteps=nsteps; }
}
////////////
final private int n;
final private int ntrials;
final private double pmin;
final private double pmax;
final private Random random = new Random();
final private Bernoulli bernoulli = new Bernoulli();
final private BitSet bits;
public EvenlySpacedOnesTest(int n, int ntrials, double pmin, double pmax) {
this.n=n; this.ntrials=ntrials; this.pmin=pmin; this.pmax=pmax;
this.bits = new BitSet(n);
}
/*
* generate random bit string
*/
private int generateBits()
{
int k = 0; // # of 1's
for (int i = 0; i < n; ++i)
{
boolean b = bernoulli.getNextBoolean();
this.bits.set(i, b);
if (b) ++k;
}
return k;
}
private int findEvenlySpacedOnes(int k, int[] pos)
{
int[] bitPosition = new int[k];
for (int i = 0, j = 0; i < n; ++i)
{
if (this.bits.get(i))
{
bitPosition[j++] = i;
}
}
int nsteps = n; // first, it takes N operations to find the bit positions.
boolean found = false;
if (k >= 3) // don't bother doing anything if there are less than 3 ones. :(
{
int lastBitSetPosition = bitPosition[k-1];
for (int j1 = 0; !found && j1 < k; ++j1)
{
pos[0] = bitPosition[j1];
for (int j2 = j1+1; !found && j2 < k; ++j2)
{
pos[1] = bitPosition[j2];
++nsteps;
pos[2] = 2*pos[1]-pos[0];
// calculate 3rd bit index that might be set;
// the other two indices point to bits that are set
if (pos[2] > lastBitSetPosition)
break;
// loop inner loop until we go out of bounds
found = this.bits.get(pos[2]);
// we're done if we find a third 1!
}
}
}
if (!found)
pos[0]=-1;
return nsteps;
}
/*
* run an algorithm that finds evenly spaced ones and returns # of steps.
*/
public TestResult run()
{
bernoulli.setProbability(pmin + (pmax-pmin)*random.nextDouble());
// probability of bernoulli process is randomly distributed between pmin and pmax
// generate bit string.
int k = generateBits();
int[] pos = new int[3];
int nsteps = findEvenlySpacedOnes(k, pos);
return new TestResult(k, nsteps);
}
public static void main(String[] args)
{
int n;
int ntrials;
double pmin = 0, pmax = 1;
try {
n = Integer.parseInt(args[0]);
ntrials = Integer.parseInt(args[1]);
if (args.length >= 3)
pmin = Double.parseDouble(args[2]);
if (args.length >= 4)
pmax = Double.parseDouble(args[3]);
}
catch (Exception e)
{
System.out.println("usage: EvenlySpacedOnesTest N NTRIALS [pmin [pmax]]");
System.exit(0);
return; // make the compiler happy
}
final StatisticalSummary[] statistics;
statistics=new StatisticalSummary[n+1];
for (int i = 0; i <= n; ++i)
{
statistics[i] = new StatisticalSummary();
}
EvenlySpacedOnesTest test = new EvenlySpacedOnesTest(n, ntrials, pmin, pmax);
int printInterval=100000;
int nextPrint = printInterval;
for (int i = 0; i < ntrials; ++i)
{
TestResult result = test.run();
statistics[result.k].add(result.nsteps);
if (i == nextPrint)
{
System.err.println(i);
nextPrint += printInterval;
}
}
StatisticalSummary.printOut(System.out, statistics);
}
}
# <algorithm>
def contains_evenly_spaced?(input)
return false if input.size < 3
one_indices = []
input.each_with_index do |digit, index|
next if digit == 0
one_indices << index
end
return false if one_indices.size < 3
previous_indexes = []
one_indices.each do |index|
if !previous_indexes.empty?
previous_indexes.each do |previous_index|
multiple = index - previous_index
success_index = index + multiple
return true if input[success_index] == 1
end
end
previous_indexes << index
end
return false
end
# </algorithm>
def parse_input(input)
input.chars.map { |c| c.to_i }
end
I'm having trouble with the worst-case scenarios with millions of digits. Fuzzing from /dev/urandom essentially gives you O(n), but I know the worst case is worse than that. I just can't tell how much worse. For small n, it's trivial to find inputs at around 3*n*log(n), but it's surprisingly hard to differentiate those from some other order of growth for this particular problem.
Can anyone who was working on worst-case inputs generate a string with length greater than say, one hundred thousand?
An adaptation of the Rabin-Karp algorithm could be possible for you.
Its complexity is 0(n) so it could help you.
Take a look http://en.wikipedia.org/wiki/Rabin-Karp_string_search_algorithm
Could this be a solution? I', not sure if it's O(nlogn) but in my opinion it's better than O(n²) because the the only way not to find a triple would be a prime number distribution.
There's room for improvement, the second found 1 could be the next first 1. Also no error checking.
#include <iostream>
#include <string>
int findIt(std::string toCheck) {
for (int i=0; i<toCheck.length(); i++) {
if (toCheck[i]=='1') {
std::cout << i << ": " << toCheck[i];
for (int j = i+1; j<toCheck.length(); j++) {
if (toCheck[j]=='1' && toCheck[(i+2*(j-i))] == '1') {
std::cout << ", " << j << ":" << toCheck[j] << ", " << (i+2*(j-i)) << ":" << toCheck[(i+2*(j-i))] << " found" << std::endl;
return 0;
}
}
}
}
return -1;
}
int main (int agrc, char* args[]) {
std::string toCheck("1001011");
findIt(toCheck);
std::cin.get();
return 0;
}
I think this algorithm has O(n log n) complexity (C++, DevStudio 2k5). Now, I don't know the details of how to analyse an algorithm to determine its complexity, so I have added some metric gathering information to the code. The code counts the number of tests done on the sequence of 1's and 0's for any given input (hopefully, I've not made a balls of the algorithm). We can compare the actual number of tests against the O value and see if there's a correlation.
#include <iostream>
using namespace std;
bool HasEvenBits (string &sequence, int &num_compares)
{
bool
has_even_bits = false;
num_compares = 0;
for (unsigned i = 1 ; i <= (sequence.length () - 1) / 2 ; ++i)
{
for (unsigned j = 0 ; j < sequence.length () - 2 * i ; ++j)
{
++num_compares;
if (sequence [j] == '1' && sequence [j + i] == '1' && sequence [j + i * 2] == '1')
{
has_even_bits = true;
// we could 'break' here, but I want to know the worst case scenario so keep going to the end
}
}
}
return has_even_bits;
}
int main ()
{
int
count;
string
input = "111";
for (int i = 3 ; i < 32 ; ++i)
{
HasEvenBits (input, count);
cout << i << ", " << count << endl;
input += "0";
}
}
This program outputs the number of tests for each string length up to 32 characters. Here's the results:
n Tests n log (n)
=====================
3 1 1.43
4 2 2.41
5 4 3.49
6 6 4.67
7 9 5.92
8 12 7.22
9 16 8.59
10 20 10.00
11 25 11.46
12 30 12.95
13 36 14.48
14 42 16.05
15 49 17.64
16 56 19.27
17 64 20.92
18 72 22.59
19 81 24.30
20 90 26.02
21 100 27.77
22 110 29.53
23 121 31.32
24 132 33.13
25 144 34.95
26 156 36.79
27 169 38.65
28 182 40.52
29 196 42.41
30 210 44.31
31 225 46.23
I've added the 'n log n' values as well. Plot these using your graphing tool of choice to see a correlation between the two results. Does this analysis extend to all values of n? I don't know.
There is known Random(0,1) function, it is a uniformed random function, which means, it will give 0 or 1, with probability 50%. Implement Random(a, b) that only makes calls to Random(0,1)
What I though so far is, put the range a-b in a 0 based array, then I have index 0, 1, 2...b-a.
then call the RANDOM(0,1) b-a times, sum the results as generated idx. and return the element.
However since there is no answer in the book, I don't know if this way is correct or the best. How to prove that the probability of returning each element is exactly same and is 1/(b-a+1) ?
And what is the right/better way to do this?
If your RANDOM(0, 1) returns either 0 or 1, each with probability 0.5 then you can generate bits until you have enough to represent the number (b-a+1) in binary. This gives you a random number in a slightly too large range: you can test and repeat if it fails. Something like this (in Python).
def rand_pow2(bit_count):
"""Return a random number with the given number of bits."""
result = 0
for i in xrange(bit_count):
result = 2 * result + RANDOM(0, 1)
return result
def random_range(a, b):
"""Return a random integer in the closed interval [a, b]."""
bit_count = math.ceil(math.log2(b - a + 1))
while True:
r = rand_pow2(bit_count)
if a + r <= b:
return a + r
When you sum random numbers, the result is not longer evenly distributed - it looks like a Gaussian function. Look up "law of large numbers" or read any probability book / article. Just like flipping coins 100 times is highly highly unlikely to give 100 heads. It's likely to give close to 50 heads and 50 tails.
Your inclination to put the range from 0 to a-b first is correct. However, you cannot do it as you stated. This question asks exactly how to do that, and the answer utilizes unique factorization. Write m=a-b in base 2, keeping track of the largest needed exponent, say e. Then, find the biggest multiple of m that is smaller than 2^e, call it k. Finally, generate e numbers with RANDOM(0,1), take them as the base 2 expansion of some number x, if x < k*m, return x, otherwise try again. The program looks something like this (simple case when m<2^2):
int RANDOM(0,m) {
// find largest power of n needed to write m in base 2
int e=0;
while (m > 2^e) {
++e;
}
// find largest multiple of m less than 2^e
int k=1;
while (k*m < 2^2) {
++k
}
--k; // we went one too far
while (1) {
// generate a random number in base 2
int x = 0;
for (int i=0; i<e; ++i) {
x = x*2 + RANDOM(0,1);
}
// if x isn't too large, return it x modulo m
if (x < m*k)
return (x % m);
}
}
Now you can simply add a to the result to get uniformly distributed numbers between a and b.
Divide and conquer could help us in generating a random number in range [a,b] using random(0,1). The idea is
if a is equal to b, then random number is a
Find mid of the range [a,b]
Generate random(0,1)
If above is 0, return a random number in range [a,mid] using recursion
else return a random number in range [mid+1, b] using recursion
The working 'C' code is as follows.
int random(int a, int b)
{
if(a == b)
return a;
int c = RANDOM(0,1); // Returns 0 or 1 with probability 0.5
int mid = a + (b-a)/2;
if(c == 0)
return random(a, mid);
else
return random(mid + 1, b);
}
If you have a RNG that returns {0, 1} with equal probability, you can easily create a RNG that returns numbers {0, 2^n} with equal probability.
To do this you just use your original RNG n times and get a binary number like 0010110111. Each of the numbers are (from 0 to 2^n) are equally likely.
Now it is easy to get a RNG from a to b, where b - a = 2^n. You just create a previous RNG and add a to it.
Now the last question is what should you do if b-a is not 2^n?
Good thing that you have to do almost nothing. Relying on rejection sampling technique. It tells you that if you have a big set and have a RNG over that set and need to select an element from a subset of this set, you can just keep selecting an element from a bigger set and discarding them till they exist in your subset.
So all you do, is find b-a and find the first n such that b-a <= 2^n. Then using rejection sampling till you picked an element smaller b-a. Than you just add a.
Suppose I have an int x = 54897, old digit index (0 based), and the new value for that digit. What's the fastest way to get the new value?
Example
x = 54897
index = 3
value = 2
y = f(x, index, value) // => 54827
Edit: by fastest, I definitely mean faster performance. No string processing.
In simplest case (considering the digits are numbered from LSB to MSB, the first one being 0) AND knowing the old digit, we could do as simple as that:
num += (new_digit - old_digit) * 10**pos;
For the real problem we would need:
1) the MSB-first version of the pos, that could cost you a log() or at most log10(MAX_INT) divisions by ten (could be improved using binary search).
2) the digit from that pos that would need at most 2 divisions (or zero, using results from step 1).
You could also use the special fpu instruction from x86 that is able to save a float in BCD (I have no idea how slow it is).
UPDATE: the first step could be done even faster, without any divisions, with a binary search like this:
int my_log10(unsigned short n){
// short: 0.. 64k -> 1.. 5 digits
if (n < 1000){ // 1..3
if (n < 10) return 1;
if (n < 100) return 2;
return 3;
} else { // 4..5
if (n < 10000) return 4;
return 5;
}
}
If your index started at the least significant digit, you could do something like
p = pow(10,index);
x = (x / (p*10) * (p*10) + value * p + x % p).
But since your index is backwards, a string is probably the way to go. It would also be more readable and maintainable.
Calculate the "mask" M: 10 raised to the power of index, where index is a zero-based index from the right. If you need to index from the left, recalculate index accordingly.
Calculate the "prefix" PRE = x / (M * 10) * (M * 10)
Calculate the "suffix" SUF = x % M
Calculate the new "middle part" MID = value * M
Generate the new number new_x = PRE + MID + POST.
P.S. ruslik's answer does it more elegantly :)
You need to start by figuring out how many digits are in your input. I can think of two ways of doing that, one with a loop and one with logarithms. Here's the loop version. This will fail for negative and zero inputs and when the index is out of bounds, probably other conditions too, but it's a starting point.
def f(x, index, value):
place = 1
residual = x
while residual > 0:
if index < 0:
place *= 10
index -= 1
residual /= 10
digit = (x / place) % 10
return x - (place * digit) + (place * value)
P.S. This is working Python code. The principle of something simple like this is easy to work out, but the details are so tricky that you really need to iterate it a bit. In this case I started with the principle that I wanted to subtract out the old digit and add the new one; from there it was a matter of getting the correct multiplier.
You gotta get specific with your compute platform if you're talking about performance.
I would approach this by converting the number into pairs of decimal digits, 4 bit each.
Then I would find and process the pair that needs modification as a byte.
Then I would put the number back together.
There are assemblers that do this very well.
I had this question on an Algorithms test yesterday, and I can't figure out the answer. It is driving me absolutely crazy, because it was worth about 40 points. I figure that most of the class didn't solve it correctly, because I haven't come up with a solution in the past 24 hours.
Given a arbitrary binary string of length n, find three evenly spaced ones within the string if they exist. Write an algorithm which solves this in O(n * log(n)) time.
So strings like these have three ones that are "evenly spaced": 11100000, 0100100100
edit: It is a random number, so it should be able to work for any number. The examples I gave were to illustrate the "evenly spaced" property. So 1001011 is a valid number. With 1, 4, and 7 being ones that are evenly spaced.
Finally! Following up leads in sdcvvc's answer, we have it: the O(n log n) algorithm for the problem! It is simple too, after you understand it. Those who guessed FFT were right.
The problem: we are given a binary string S of length n, and we want to find three evenly spaced 1s in it. For example, S may be 110110010, where n=9. It has evenly spaced 1s at positions 2, 5, and 8.
Scan S left to right, and make a list L of positions of 1. For the S=110110010 above, we have the list L = [1, 2, 4, 5, 8]. This step is O(n). The problem is now to find an arithmetic progression of length 3 in L, i.e. to find distinct a, b, c in L such that b-a = c-b, or equivalently a+c=2b. For the example above, we want to find the progression (2, 5, 8).
Make a polynomial p with terms xk for each k in L. For the example above, we make the polynomial p(x) = (x + x2 + x4 + x5+x8). This step is O(n).
Find the polynomial q = p2, using the Fast Fourier Transform. For the example above, we get the polynomial q(x) = x16 + 2x13 + 2x12 + 3x10 + 4x9 + x8 + 2x7 + 4x6 + 2x5 + x4 + 2x3 + x2. This step is O(n log n).
Ignore all terms except those corresponding to x2k for some k in L. For the example above, we get the terms x16, 3x10, x8, x4, x2. This step is O(n), if you choose to do it at all.
Here's the crucial point: the coefficient of any x2b for b in L is precisely the number of pairs (a,c) in L such that a+c=2b. [CLRS, Ex. 30.1-7] One such pair is (b,b) always (so the coefficient is at least 1), but if there exists any other pair (a,c), then the coefficient is at least 3, from (a,c) and (c,a). For the example above, we have the coefficient of x10 to be 3 precisely because of the AP (2,5,8). (These coefficients x2b will always be odd numbers, for the reasons above. And all other coefficients in q will always be even.)
So then, the algorithm is to look at the coefficients of these terms x2b, and see if any of them is greater than 1. If there is none, then there are no evenly spaced 1s. If there is a b in L for which the coefficient of x2b is greater than 1, then we know that there is some pair (a,c) — other than (b,b) — for which a+c=2b. To find the actual pair, we simply try each a in L (the corresponding c would be 2b-a) and see if there is a 1 at position 2b-a in S. This step is O(n).
That's all, folks.
One might ask: do we need to use FFT? Many answers, such as beta's, flybywire's, and rsp's, suggest that the approach that checks each pair of 1s and sees if there is a 1 at the "third" position, might work in O(n log n), based on the intuition that if there are too many 1s, we would find a triple easily, and if there are too few 1s, checking all pairs takes little time. Unfortunately, while this intuition is correct and the simple approach is better than O(n2), it is not significantly better. As in sdcvvc's answer, we can take the "Cantor-like set" of strings of length n=3k, with 1s at the positions whose ternary representation has only 0s and 2s (no 1s) in it. Such a string has 2k = n(log 2)/(log 3) ≈ n0.63 ones in it and no evenly spaced 1s, so checking all pairs would be of the order of the square of the number of 1s in it: that's 4k ≈ n1.26 which unfortunately is asymptotically much larger than (n log n). In fact, the worst case is even worse: Leo Moser in 1953 constructed (effectively) such strings which have n1-c/√(log n) 1s in them but no evenly spaced 1s, which means that on such strings, the simple approach would take Θ(n2-2c/√(log n)) — only a tiny bit better than Θ(n2), surprisingly!
About the maximum number of 1s in a string of length n with no 3 evenly spaced ones (which we saw above was at least n0.63 from the easy Cantor-like construction, and at least n1-c/√(log n) with Moser's construction) — this is OEIS A003002. It can also be calculated directly from OEIS A065825 as the k such that A065825(k) ≤ n < A065825(k+1). I wrote a program to find these, and it turns out that the greedy algorithm does not give the longest such string. For example, for n=9, we can get 5 1s (110100011) but the greedy gives only 4 (110110000), for n=26 we can get 11 1s (11001010001000010110001101) but the greedy gives only 8 (11011000011011000000000000), and for n=74 we can get 22 1s (11000010110001000001011010001000000000000000010001011010000010001101000011) but the greedy gives only 16 (11011000011011000000000000011011000011011000000000000000000000000000000000). They do agree at quite a few places until 50 (e.g. all of 38 to 50), though. As the OEIS references say, it seems that Jaroslaw Wroblewski is interested in this question, and he maintains a website on these non-averaging sets. The exact numbers are known only up to 194.
Your problem is called AVERAGE in this paper (1999):
A problem is 3SUM-hard if there is a sub-quadratic reduction from the problem 3SUM: Given a set A of n integers, are there elements a,b,c in A such that a+b+c = 0? It is not known whether AVERAGE is 3SUM-hard. However, there is a simple linear-time reduction from AVERAGE to 3SUM, whose description we omit.
Wikipedia:
When the integers are in the range [−u ... u], 3SUM can be solved in time O(n + u lg u) by representing S as a bit vector and performing a convolution using FFT.
This is enough to solve your problem :).
What is very important is that O(n log n) is complexity in terms of number of zeroes and ones, not the count of ones (which could be given as an array, like [1,5,9,15]). Checking if a set has an arithmetic progression, terms of number of 1's, is hard, and according to that paper as of 1999 no faster algorithm than O(n2) is known, and is conjectured that it doesn't exist. Everybody who doesn't take this into account is attempting to solve an open problem.
Other interesting info, mostly irrevelant:
Lower bound:
An easy lower bound is Cantor-like set (numbers 1..3^n-1 not containing 1 in their ternary expansion) - its density is n^(log_3 2) (circa 0.631). So any checking if the set isn't too large, and then checking all pairs is not enough to get O(n log n). You have to investigate the sequence smarter. A better lower bound is quoted here - it's n1-c/(log(n))^(1/2). This means Cantor set is not optimal.
Upper bound - my old algorithm:
It is known that for large n, a subset of {1,2,...,n} not containing arithmetic progression has at most n/(log n)^(1/20) elements. The paper On triples in arithmetic progression proves more: the set cannot contain more than n * 228 * (log log n / log n)1/2 elements. So you could check if that bound is achieved and if not, naively check pairs. This is O(n2 * log log n / log n) algorithm, faster than O(n2). Unfortunately "On triples..." is on Springer - but the first page is available, and Ben Green's exposition is available here, page 28, theorem 24.
By the way, the papers are from 1999 - the same year as the first one I mentioned, so that's probably why the first one doesn't mention that result.
This is not a solution, but a similar line of thought to what Olexiy was thinking
I was playing around with creating sequences with maximum number of ones, and they are all quite interesting, I got up to 125 digits and here are the first 3 numbers it found by attempting to insert as many '1' bits as possible:
11011000011011000000000000001101100001101100000000000000000000000000000000000000000110110000110110000000000000011011000011011
10110100010110100000000000010110100010110100000000000000000000000000000000000000000101101000101101000000000000101101000101101
10011001010011001000000000010011001010011001000000000000000000000000000000000000010011001010011001000000000010011001010011001
Notice they are all fractals (not too surprising given the constraints). There may be something in thinking backwards, perhaps if the string is not a fractal of with a characteristic, then it must have a repeating pattern?
Thanks to beta for the better term to describe these numbers.
Update:
Alas it looks like the pattern breaks down when starting with a large enough initial string, such as: 10000000000001:
100000000000011
10000000000001101
100000000000011011
10000000000001101100001
100000000000011011000011
10000000000001101100001101
100000000000011011000011010000000001
100000000000011011000011010000000001001
1000000000000110110000110100000000010011
1000000000000110110000110100000000010011001
10000000000001101100001101000000000100110010000000001
10000000000001101100001101000000000100110010000000001000001
1000000000000110110000110100000000010011001000000000100000100000000000001
10000000000001101100001101000000000100110010000000001000001000000000000011
1000000000000110110000110100000000010011001000000000100000100000000000001101
100000000000011011000011010000000001001100100000000010000010000000000000110100001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001
1000000000000110110000110100000000010011001000000000100000100000000000001101000010010000010000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001
1000000000000110110000110100000000010011001000000000100000100000000000001101000010010000010000001100010000000010000000000000000000000000000000000000000100000010000000000000011
1000000000000110110000110100000000010011001000000000100000100000000000001101000010010000010000001100010000000010000000000000000000000000000000000000000100000010000000000000011000000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001001000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001001000100100000100000000000001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001001000001000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001001000100100000100000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001001000001000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000011
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001001000001000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000011000001
1000000000000110110000110100000000010011001000000000100000100000000000001101000010010000010000001100010000000010000000000000000000000000000000000000000100000010000000000000011000000001100100000000100100000000000010000000010000100000100100010010000010000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000110000010000000000000000000001
1000000000000110110000110100000000010011001000000000100000100000000000001101000010010000010000001100010000000010000000000000000000000000000000000000000100000010000000000000011000000001100100000000100100000000000010000000010000100000100100010010000010000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000110000010000000000000000000001001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001001000100100000100000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000001100000100000000000000000000010010000000000000000000000000000000000001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001001000001000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000011000001000000000000000000000100100000000000000000000000000000000000011
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001001000001000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000011000001000000000000000000000100100000000000000000000000000000000000011001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001001000100100000100000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000001100000100000000000000000000010010000000000000000000000000000000000001100100000000000000000000001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001001000100100000100000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000001100000100000000000000000000010010000000000000000000000000000000000001100100000000000000000000001001
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001001000100100000100000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000001100000100000000000000000000010010000000000000000000000000000000000001100100000000000000000000001001000001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001000000110001000000001000000000000000000000000000000000000000010000001000000000000001100000000110010000000010010000000000001000000001000010000010010001001000001000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000011000001000000000000000000000100100000000000000000000000000000000000011001000000000000000000000010010000010000001
1000000000000110110000110100000000010011001000000000100000100000000000001101000010010000010000001100010000000010000000000000000000000000000000000000000100000010000000000000011000000001100100000000100100000000000010000000010000100000100100010010000010000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000110000010000000000000000000001001000000000000000000000000000000000000110010000000000000000000000100100000100000011
10000000000001101100001101000000000100110010000000001000001000000000000011010000100100000100000011000100000000100000000000000000000000000000000000000001000000100000000000000110000000011001000000001001000000000000100000000100001000001001000100100000100000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000001100000100000000000000000000010010000000000000000000000000000000000001100100000000000000000000001001000001000000110000000000001
I suspect that a simple approach that looks like O(n^2) will actually yield something better, like O(n ln(n)). The sequences that take the longest to test (for any given n) are the ones that contain no trios, and that puts severe restrictions on the number of 1's that can be in the sequence.
I've come up with some hand-waving arguments, but I haven't been able to find a tidy proof. I'm going to take a stab in the dark: the answer is a very clever idea that the professor has known for so long that it's come to seem obvious, but it's much too hard for the students. (Either that or you slept through the lecture that covered it.)
Revision: 2009-10-17 23:00
I've run this on large numbers (like, strings of 20 million) and I now believe this algorithm is not O(n logn). Notwithstanding that, it's a cool enough implementation and contains a number of optimizations that makes it run really fast. It evaluates all the arrangements of binary strings 24 or fewer digits in under 25 seconds.
I've updated the code to include the 0 <= L < M < U <= X-1 observation from earlier today.
Original
This is, in concept, similar to another question I answered. That code also looked at three values in a series and determined if a triplet satisfied a condition. Here is C# code adapted from that:
using System;
using System.Collections.Generic;
namespace StackOverflow1560523
{
class Program
{
public struct Pair<T>
{
public T Low, High;
}
static bool FindCandidate(int candidate,
List<int> arr,
List<int> pool,
Pair<int> pair,
ref int iterations)
{
int lower = pair.Low, upper = pair.High;
while ((lower >= 0) && (upper < pool.Count))
{
int lowRange = candidate - arr[pool[lower]];
int highRange = arr[pool[upper]] - candidate;
iterations++;
if (lowRange < highRange)
lower -= 1;
else if (lowRange > highRange)
upper += 1;
else
return true;
}
return false;
}
static List<int> BuildOnesArray(string s)
{
List<int> arr = new List<int>();
for (int i = 0; i < s.Length; i++)
if (s[i] == '1')
arr.Add(i);
return arr;
}
static void BuildIndexes(List<int> arr,
ref List<int> even, ref List<int> odd,
ref List<Pair<int>> evenIndex, ref List<Pair<int>> oddIndex)
{
for (int i = 0; i < arr.Count; i++)
{
bool isEven = (arr[i] & 1) == 0;
if (isEven)
{
evenIndex.Add(new Pair<int> {Low=even.Count-1, High=even.Count+1});
oddIndex.Add(new Pair<int> {Low=odd.Count-1, High=odd.Count});
even.Add(i);
}
else
{
oddIndex.Add(new Pair<int> {Low=odd.Count-1, High=odd.Count+1});
evenIndex.Add(new Pair<int> {Low=even.Count-1, High=even.Count});
odd.Add(i);
}
}
}
static int FindSpacedOnes(string s)
{
// List of indexes of 1s in the string
List<int> arr = BuildOnesArray(s);
//if (s.Length < 3)
// return 0;
// List of indexes to odd indexes in arr
List<int> odd = new List<int>(), even = new List<int>();
// evenIndex has indexes into arr to bracket even numbers
// oddIndex has indexes into arr to bracket odd numbers
List<Pair<int>> evenIndex = new List<Pair<int>>(),
oddIndex = new List<Pair<int>>();
BuildIndexes(arr,
ref even, ref odd,
ref evenIndex, ref oddIndex);
int iterations = 0;
for (int i = 1; i < arr.Count-1; i++)
{
int target = arr[i];
bool found = FindCandidate(target, arr, odd, oddIndex[i], ref iterations) ||
FindCandidate(target, arr, even, evenIndex[i], ref iterations);
if (found)
return iterations;
}
return iterations;
}
static IEnumerable<string> PowerSet(int n)
{
for (long i = (1L << (n-1)); i < (1L << n); i++)
{
yield return Convert.ToString(i, 2).PadLeft(n, '0');
}
}
static void Main(string[] args)
{
for (int i = 5; i < 64; i++)
{
int c = 0;
string hardest_string = "";
foreach (string s in PowerSet(i))
{
int cost = find_spaced_ones(s);
if (cost > c)
{
hardest_string = s;
c = cost;
Console.Write("{0} {1} {2}\r", i, c, hardest_string);
}
}
Console.WriteLine("{0} {1} {2}", i, c, hardest_string);
}
}
}
}
The principal differences are:
Exhaustive search of solutions
This code generates a power set of data to find the hardest input to solve for this algorithm.
All solutions versus hardest to solve
The code for the previous question generated all the solutions using a python generator. This code just displays the hardest for each pattern length.
Scoring algorithm
This code checks the distance from the middle element to its left- and right-hand edge. The python code tested whether a sum was above or below 0.
Convergence on a candidate
The current code works from the middle towards the edge to find a candidate. The code in the previous problem worked from the edges towards the middle. This last change gives a large performance improvement.
Use of even and odd pools
Based on the observations at the end of this write-up, the code searches pairs of even numbers of pairs of odd numbers to find L and U, keeping M fixed. This reduces the number of searches by pre-computing information. Accordingly, the code uses two levels of indirection in the main loop of FindCandidate and requires two calls to FindCandidate for each middle element: once for even numbers and once for odd ones.
The general idea is to work on indexes, not the raw representation of the data. Calculating an array where the 1's appear allows the algorithm to run in time proportional to the number of 1's in the data rather than in time proportional to the length of the data. This is a standard transformation: create a data structure that allows faster operation while keeping the problem equivalent.
The results are out of date: removed.
Edit: 2009-10-16 18:48
On yx's data, which is given some credence in the other responses as representative of hard data to calculate on, I get these results... I removed these. They are out of date.
I would point out that this data is not the hardest for my algorithm, so I think the assumption that yx's fractals are the hardest to solve is mistaken. The worst case for a particular algorithm, I expect, will depend upon the algorithm itself and will not likely be consistent across different algorithms.
Edit: 2009-10-17 13:30
Further observations on this.
First, convert the string of 0's and 1's into an array of indexes for each position of the 1's. Say the length of that array A is X. Then the goal is to find
0 <= L < M < U <= X-1
such that
A[M] - A[L] = A[U] - A[M]
or
2*A[M] = A[L] + A[U]
Since A[L] and A[U] sum to an even number, they can't be (even, odd) or (odd, even). The search for a match could be improved by splitting A[] into odd and even pools and searching for matches on A[M] in the pools of odd and even candidates in turn.
However, this is more of a performance optimization than an algorithmic improvement, I think. The number of comparisons should drop, but the order of the algorithm should be the same.
Edit 2009-10-18 00:45
Yet another optimization occurs to me, in the same vein as separating the candidates into even and odd. Since the three indexes have to add to a multiple of 3 (a, a+x, a+2x -- mod 3 is 0, regardless of a and x), you can separate L, M, and U into their mod 3 values:
M L U
0 0 0
1 2
2 1
1 0 2
1 1
2 0
2 0 1
1 0
2 2
In fact, you could combine this with the even/odd observation and separate them into their mod 6 values:
M L U
0 0 0
1 5
2 4
3 3
4 2
5 1
and so on. This would provide a further performance optimization but not an algorithmic speedup.
Wasn't able to come up with the solution yet :(, but have some ideas.
What if we start from a reverse problem: construct a sequence with the maximum number of 1s and WITHOUT any evenly spaced trios. If you can prove the maximum number of 1s is o(n), then you can improve your estimate by iterating only through list of 1s only.
This may help....
This problem reduces to the following:
Given a sequence of positive integers, find a contiguous subsequence partitioned into a prefix and a suffix such that the sum of the prefix of the subsequence is equal to the sum of the suffix of the subsequence.
For example, given a sequence of [ 3, 5, 1, 3, 6, 5, 2, 2, 3, 5, 6, 4 ], we would find a subsequence of [ 3, 6, 5, 2, 2] with a prefix of [ 3, 6 ] with prefix sum of 9 and a suffix of [ 5, 2, 2 ] with suffix sum of 9.
The reduction is as follows:
Given a sequence of zeros and ones, and starting at the leftmost one, continue moving to the right. Each time another one is encountered, record the number of moves since the previous one was encountered and append that number to the resulting sequence.
For example, given a sequence of [ 0, 1, 1, 0, 0, 1, 0, 0, 0, 1 0 ], we would find the reduction of [ 1, 3, 4]. From this reduction, we calculate the contiguous subsequence of [ 1, 3, 4], the prefix of [ 1, 3] with sum of 4, and the suffix of [ 4 ] with sum of 4.
This reduction may be computed in O(n).
Unfortunately, I am not sure where to go from here.
For the simple problem type (i.e. you search three "1" with only (i.e. zero or more) "0" between it), Its quite simple: You could just split the sequence at every "1" and look for two adjacent subsequences having the same length (the second subsequence not being the last one, of course). Obviously, this can be done in O(n) time.
For the more complex version (i.e. you search an index i and an gap g>0 such that s[i]==s[i+g]==s[i+2*g]=="1"), I'm not sure, if there exists an O(n log n) solution, since there are possibly O(n²) triplets having this property (think of a string of all ones, there are approximately n²/2 such triplets). Of course, you are looking for only one of these, but I have currently no idea, how to find it ...
A fun question, but once you realise that the actual pattern between two '1's does not matter, the algorithm becomes:
scan look for a '1'
starting from the next position scan for another '1' (to the end of the array minus the distance from the current first '1' or else the 3rd '1' would be out of bounds)
if at the position of the 2nd '1' plus the distance to the first 1' a third '1' is found, we have evenly spaces ones.
In code, JTest fashion, (Note this code isn't written to be most efficient and I added some println's to see what happens.)
import java.util.Random;
import junit.framework.TestCase;
public class AlgorithmTest extends TestCase {
/**
* Constructor for GetNumberTest.
*
* #param name The test's name.
*/
public AlgorithmTest(String name) {
super(name);
}
/**
* #see TestCase#setUp()
*/
protected void setUp() throws Exception {
super.setUp();
}
/**
* #see TestCase#tearDown()
*/
protected void tearDown() throws Exception {
super.tearDown();
}
/**
* Tests the algorithm.
*/
public void testEvenlySpacedOnes() {
assertFalse(isEvenlySpaced(1));
assertFalse(isEvenlySpaced(0x058003));
assertTrue(isEvenlySpaced(0x07001));
assertTrue(isEvenlySpaced(0x01007));
assertTrue(isEvenlySpaced(0x101010));
// some fun tests
Random random = new Random();
isEvenlySpaced(random.nextLong());
isEvenlySpaced(random.nextLong());
isEvenlySpaced(random.nextLong());
}
/**
* #param testBits
*/
private boolean isEvenlySpaced(long testBits) {
String testString = Long.toBinaryString(testBits);
char[] ones = testString.toCharArray();
final char ONE = '1';
for (int n = 0; n < ones.length - 1; n++) {
if (ONE == ones[n]) {
for (int m = n + 1; m < ones.length - m + n; m++) {
if (ONE == ones[m] && ONE == ones[m + m - n]) {
System.out.println(" IS evenly spaced: " + testBits + '=' + testString);
System.out.println(" at: " + n + ", " + m + ", " + (m + m - n));
return true;
}
}
}
}
System.out.println("NOT evenly spaced: " + testBits + '=' + testString);
return false;
}
}
I thought of a divide-and-conquer approach that might work.
First, in preprocessing you need to insert all numbers less than one half your input size (n/3) into a list.
Given a string: 0000010101000100 (note that this particular example is valid)
Insert all primes (and 1) from 1 to (16/2) into a list: {1, 2, 3, 4, 5, 6, 7}
Then divide it in half:
100000101 01000100
Keep doing this until you get to strings of size 1. For all size-one strings with a 1 in them, add the index of the string to the list of possibilities; otherwise, return -1 for failure.
You'll also need to return a list of still-possible spacing distances, associated with each starting index. (Start with the list you made above and remove numbers as you go) Here, an empty list means you're only dealing with one 1 and so any spacing is possible at this point; otherwise the list includes spacings that must be ruled out.
So continuing with the example above:
1000 0101 0100 0100
10 00 01 01 01 00 01 00
1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0
In the first combine step, we have eight sets of two now. In the first, we have the possibility of a set, but we learn that spacing by 1 is impossible because of the other zero being there. So we return 0 (for the index) and {2,3,4,5,7} for the fact that spacing by 1 is impossible. In the second, we have nothing and so return -1. In the third we have a match with no spacings eliminated in index 5, so return 5, {1,2,3,4,5,7}. In the fourth pair we return 7, {1,2,3,4,5,7}. In the fifth, return 9, {1,2,3,4,5,7}. In the sixth, return -1. In the seventh, return 13, {1,2,3,4,5,7}. In the eighth, return -1.
Combining again into four sets of four, we have:
1000: Return (0, {4,5,6,7})
0101: Return (5, {2,3,4,5,6,7}), (7, {1,2,3,4,5,6,7})
0100: Return (9, {3,4,5,6,7})
0100: Return (13, {3,4,5,6,7})
Combining into sets of eight:
10000101: Return (0, {5,7}), (5, {2,3,4,5,6,7}), (7, {1,2,3,4,5,6,7})
01000100: Return (9, {4,7}), (13, {3,4,5,6,7})
Combining into a set of sixteen:
10000101 01000100
As we've progressed, we keep checking all the possibilities so far. Up to this step we've left stuff that went beyond the end of the string, but now we can check all the possibilities.
Basically, we check the first 1 with spacings of 5 and 7, and find that they don't line up to 1's. (Note that each check is CONSTANT, not linear time) Then we check the second one (index 5) with spacings of 2, 3, 4, 5, 6, and 7-- or we would, but we can stop at 2 since that actually matches up.
Phew! That's a rather long algorithm.
I don't know 100% if it's O(n log n) because of the last step, but everything up to there is definitely O(n log n) as far as I can tell. I'll get back to this later and try to refine the last step.
EDIT: Changed my answer to reflect Welbog's comment. Sorry for the error. I'll write some pseudocode later, too, when I get a little more time to decipher what I wrote again. ;-)
I'll give my rough guess here, and let those who are better with calculating complexity to help me on how my algorithm fares in O-notation wise
given binary string 0000010101000100 (as example)
crop head and tail of zeroes -> 00000 101010001 00
we get 101010001 from previous calculation
check if the middle bit is 'one', if true, found valid three evenly spaced 'ones' (only if the number of bits is odd numbered)
correlatively, if the remained cropped number of bits is even numbered, the head and tail 'one' cannot be part of evenly spaced 'one',
we use 1010100001 as example (with an extra 'zero' to become even numbered crop), in this case we need to crop again, then becomes -> 10101 00001
we get 10101 from previous calculation, and check middle bit, and we found the evenly spaced bit again
I have no idea how to calculate complexity for this, can anyone help?
edit: add some code to illustrate my idea
edit2: tried to compile my code and found some major mistakes, fixed
char *binaryStr = "0000010101000100";
int main() {
int head, tail, pos;
head = 0;
tail = strlen(binaryStr)-1;
if( (pos = find3even(head, tail)) >=0 )
printf("found it at position %d\n", pos);
return 0;
}
int find3even(int head, int tail) {
int pos = 0;
if(head >= tail) return -1;
while(binaryStr[head] == '0')
if(head<tail) head++;
while(binaryStr[tail] == '0')
if(head<tail) tail--;
if(head >= tail) return -1;
if( (tail-head)%2 == 0 && //true if odd numbered
(binaryStr[head + (tail-head)/2] == '1') ) {
return head;
}else {
if( (pos = find3even(head, tail-1)) >=0 )
return pos;
if( (pos = find3even(head+1, tail)) >=0 )
return pos;
}
return -1;
}
I came up with something like this:
def IsSymetric(number):
number = number.strip('0')
if len(number) < 3:
return False
if len(number) % 2 == 0:
return IsSymetric(number[1:]) or IsSymetric(number[0:len(number)-2])
else:
if number[len(number)//2] == '1':
return True
return IsSymetric(number[:(len(number)//2)]) or IsSymetric(number[len(number)//2+1:])
return False
This is inspired by andycjw.
Truncate the zeros.
If even then test two substring 0 - (len-2) (skip last character) and from 1 - (len-1) (skip the first char)
If not even than if the middle char is one than we have success. Else divide the string in the midle without the midle element and check both parts.
As to the complexity this might be O(nlogn) as in each recursion we are dividing by two.
Hope it helps.
Ok, I'm going to take another stab at the problem. I think I can prove a O(n log(n)) algorithm that is similar to those already discussed by using a balanced binary tree to store distances between 1's. This approach was inspired by Justice's observation about reducing the problem to a list of distances between the 1's.
Could we scan the input string to construct a balanced binary tree around the position of 1's such that each node stores the position of the 1 and each edge is labeled with the distance to the adjacent 1 for each child node. For example:
10010001 gives the following tree
3
/ \
2 / \ 3
/ \
0 7
This can be done in O(n log(n)) since, for a string of size n, each insertion takes O(log(n)) in the worst case.
Then the problem is to search the tree to discover whether, at any node, there is a path from that node through the left-child that has the same distance as a path through the right child. This can be done recursively on each subtree. When merging two subtrees in the search, we must compare the distances from paths in the left subtree with distances from paths in the right. Since the number of paths in a subtree will be proportional to log(n), and the number of nodes is n, I believe this can be done in O(n log(n)) time.
Did I miss anything?
This seemed liked a fun problem so I decided to try my hand at it.
I am making the assumption that 111000001 would find the first 3 ones and be successful. Essentially the number of zeroes following the 1 is the important thing, since 0111000 is the same as 111000 according to your definition. Once you find two cases of 1, the next 1 found completes the trilogy.
Here it is in Python:
def find_three(bstring):
print bstring
dict = {}
lastone = -1
zerocount = 0
for i in range(len(bstring)):
if bstring[i] == '1':
print i, ': 1'
if lastone != -1:
if(zerocount in dict):
dict[zerocount].append(lastone)
if len(dict[zerocount]) == 2:
dict[zerocount].append(i)
return True, dict
else:
dict[zerocount] = [lastone]
lastone = i
zerocount = 0
else:
zerocount = zerocount + 1
#this is really just book keeping, as we have failed at this point
if lastone != -1:
if(zerocount in dict):
dict[zerocount].append(lastone)
else:
dict[zerocount] = [lastone]
return False, dict
This is a first try, so I'm sure this could be written in a cleaner manner. Please list the cases where this method fails down below.
I assume the reason this is nlog(n) is due to the following:
To find the 1 that is the start of the triplet, you need to check (n-2) characters. If you haven't found it by that point, you won't (chars n-1 and n cannot start a triplet) (O(n))
To find the second 1 that is the part of the triplet (started by the first one), you need to check m/2 (m=n-x, where x is the offset of the first 1) characters. This is because, if you haven't found the second 1 by the time you're halfway from the first one to the end, you won't... since the third 1 must be exactly the same distance past the second. (O(log(n)))
It O(1) to find the last 1 since you know the index it must be at by the time you find the first and second.
So, you have n, log(n), and 1... O(nlogn)
Edit: Oops, my bad. My brain had it set that n/2 was logn... which it obviously isn't (doubling the number on items still doubles the number of iterations on the inner loop). This is still at n^2, not solving the problem. Well, at least I got to write some code :)
Implementation in Tcl
proc get-triplet {input} {
for {set first 0} {$first < [string length $input]-2} {incr first} {
if {[string index $input $first] != 1} {
continue
}
set start [expr {$first + 1}]
set end [expr {1+ $first + (([string length $input] - $first) /2)}]
for {set second $start} {$second < $end} {incr second} {
if {[string index $input $second] != 1} {
continue
}
set last [expr {($second - $first) + $second}]
if {[string index $input $last] == 1} {
return [list $first $second $last]
}
}
}
return {}
}
get-triplet 10101 ;# 0 2 4
get-triplet 10111 ;# 0 2 4
get-triplet 11100000 ;# 0 1 2
get-triplet 0100100100 ;# 1 4 7
I think I have found a way of solving the problem, but I can't construct a formal proof. The solution I made is written in Java, and it uses a counter 'n' to count how many list/array accesses it does. So n should be less than or equal to stringLength*log(stringLength) if it is correct. I tried it for the numbers 0 to 2^22, and it works.
It starts by iterating over the input string and making a list of all the indexes which hold a one. This is just O(n).
Then from the list of indexes it picks a firstIndex, and a secondIndex which is greater than the first. These two indexes must hold ones, because they are in the list of indexes. From there the thirdIndex can be calculated. If the inputString[thirdIndex] is a 1 then it halts.
public static int testString(String input){
//n is the number of array/list accesses in the algorithm
int n=0;
//Put the indices of all the ones into a list, O(n)
ArrayList<Integer> ones = new ArrayList<Integer>();
for(int i=0;i<input.length();i++){
if(input.charAt(i)=='1'){
ones.add(i);
}
}
//If less than three ones in list, just stop
if(ones.size()<3){
return n;
}
int firstIndex, secondIndex, thirdIndex;
for(int x=0;x<ones.size()-2;x++){
n++;
firstIndex = ones.get(x);
for(int y=x+1; y<ones.size()-1; y++){
n++;
secondIndex = ones.get(y);
thirdIndex = secondIndex*2 - firstIndex;
if(thirdIndex >= input.length()){
break;
}
n++;
if(input.charAt(thirdIndex) == '1'){
//This case is satisfied if it has found three evenly spaced ones
//System.out.println("This one => " + input);
return n;
}
}
}
return n;
}
additional note: the counter n is not incremented when it iterates over the input string to construct the list of indexes. This operation is O(n), so it won't have an effect on the algorithm complexity anyway.
One inroad into the problem is to think of factors and shifting.
With shifting, you compare the string of ones and zeroes with a shifted version of itself. You then take matching ones. Take this example shifted by two:
1010101010
1010101010
------------
001010101000
The resulting 1's (bitwise ANDed), must represent all those 1's which are evenly spaced by two. The same example shifted by three:
1010101010
1010101010
-------------
0000000000000
In this case there are no 1's which are evenly spaced three apart.
So what does this tell you? Well that you only need to test shifts which are prime numbers. For example say you have two 1's which are six apart. You would only have to test 'two' shifts and 'three' shifts (since these divide six). For example:
10000010
10000010 (Shift by two)
10000010
10000010 (We have a match)
10000010
10000010 (Shift by three)
10000010 (We have a match)
So the only shifts you ever need to check are 2,3,5,7,11,13 etc. Up to the prime closest to the square root of size of the string of digits.
Nearly solved?
I think I am closer to a solution. Basically:
Scan the string for 1's. For each 1 note it's remainder after taking a modulus of its position. The modulus ranges from 1 to half the size of the string. This is because the largest possible separation size is half the string. This is done in O(n^2). BUT. Only prime moduli need be checked so O(n^2/log(n))
Sort the list of modulus/remainders in order largest modulus first, this can be done in O(n*log(n)) time.
Look for three consecutive moduli/remainders which are the same.
Somehow retrieve the position of the ones!
I think the biggest clue to the answer, is that the fastest sort algorithms, are O(n*log(n)).
WRONG
Step 1 is wrong as pointed out by a colleague. If we have 1's at position 2,12 and 102. Then taking a modulus of 10, they would all have the same remainders, and yet are not equally spaced apart! Sorry.
Here are some thoughts that, despite my best efforts, will not seem to wrap themselves up in a bow. Still, they might be a useful starting point for someone's analysis.
Consider the proposed solution as follows, which is the approach that several folks have suggested, including myself in a prior version of this answer. :)
Trim leading and trailing zeroes.
Scan the string looking for 1's.
When a 1 is found:
Assume that it is the middle 1 of the solution.
For each prior 1, use its saved position to compute the anticipated position of the final 1.
If the computed position is after the end of the string it cannot be part of the solution, so drop the position from the list of candidates.
Check the solution.
If the solution was not found, add the current 1 to the list of candidates.
Repeat until no more 1's are found.
Now consider input strings strings like the following, which will not have a solution:
101
101001
1010010001
101001000100001
101001000100001000001
In general, this is the concatenation of k strings of the form j 0's followed by a 1 for j from zero to k-1.
k=2 101
k=3 101001
k=4 1010010001
k=5 101001000100001
k=6 101001000100001000001
Note that the lengths of the substrings are 1, 2, 3, etc. So, problem size n has substrings of lengths 1 to k such that n = k(k+1)/2.
k=2 n= 3 101
k=3 n= 6 101001
k=4 n=10 1010010001
k=5 n=15 101001000100001
k=6 n=21 101001000100001000001
Note that k also tracks the number of 1's that we have to consider. Remember that every time we see a 1, we need to consider all the 1's seen so far. So when we see the second 1, we only consider the first, when we see the third 1, we reconsider the first two, when we see the fourth 1, we need to reconsider the first three, and so on. By the end of the algorithm, we've considered k(k-1)/2 pairs of 1's. Call that p.
k=2 n= 3 p= 1 101
k=3 n= 6 p= 3 101001
k=4 n=10 p= 6 1010010001
k=5 n=15 p=10 101001000100001
k=6 n=21 p=15 101001000100001000001
The relationship between n and p is that n = p + k.
The process of going through the string takes O(n) time. Each time a 1 is encountered, a maximum of (k-1) comparisons are done. Since n = k(k+1)/2, n > k**2, so sqrt(n) > k. This gives us O(n sqrt(n)) or O(n**3/2). Note however that may not be a really tight bound, because the number of comparisons goes from 1 to a maximum of k, it isn't k the whole time. But I'm not sure how to account for that in the math.
It still isn't O(n log(n)). Also, I can't prove those inputs are the worst cases, although I suspect they are. I think a denser packing of 1's to the front results in an even sparser packing at the end.
Since someone may still find it useful, here's my code for that solution in Perl:
#!/usr/bin/perl
# read input as first argument
my $s = $ARGV[0];
# validate the input
$s =~ /^[01]+$/ or die "invalid input string\n";
# strip leading and trailing 0's
$s =~ s/^0+//;
$s =~ s/0+$//;
# prime the position list with the first '1' at position 0
my #p = (0);
# start at position 1, which is the second character
my $i = 1;
print "the string is $s\n\n";
while ($i < length($s)) {
if (substr($s, $i, 1) eq '1') {
print "found '1' at position $i\n";
my #t = ();
# assuming this is the middle '1', go through the positions
# of all the prior '1's and check whether there's another '1'
# in the correct position after this '1' to make a solution
while (scalar #p) {
# $p is the position of the prior '1'
my $p = shift #p;
# $j is the corresponding position for the following '1'
my $j = 2 * $i - $p;
# if $j is off the end of the string then we don't need to
# check $p anymore
next if ($j >= length($s));
print "checking positions $p, $i, $j\n";
if (substr($s, $j, 1) eq '1') {
print "\nsolution found at positions $p, $i, $j\n";
exit 0;
}
# if $j isn't off the end of the string, keep $p for next time
push #t, $p;
}
#p = #t;
# add this '1' to the list of '1' positions
push #p, $i;
}
$i++;
}
print "\nno solution found\n";
While scanning 1s, add their positions to a List. When adding the second and successive 1s, compare them to each position in the list so far. Spacing equals currentOne (center) - previousOne (left). The right-side bit is currentOne + spacing. If it's 1, the end.
The list of ones grows inversely with the space between them. Simply stated, if you've got a lot of 0s between the 1s (as in a worst case), your list of known 1s will grow quite slowly.
using System;
using System.Collections.Generic;
namespace spacedOnes
{
class Program
{
static int[] _bits = new int[8] {128, 64, 32, 16, 8, 4, 2, 1};
static void Main(string[] args)
{
var bytes = new byte[4];
var r = new Random();
r.NextBytes(bytes);
foreach (var b in bytes) {
Console.Write(getByteString(b));
}
Console.WriteLine();
var bitCount = bytes.Length * 8;
var done = false;
var onePositions = new List<int>();
for (var i = 0; i < bitCount; i++)
{
if (isOne(bytes, i)) {
if (onePositions.Count > 0) {
foreach (var knownOne in onePositions) {
var spacing = i - knownOne;
var k = i + spacing;
if (k < bitCount && isOne(bytes, k)) {
Console.WriteLine("^".PadLeft(knownOne + 1) + "^".PadLeft(spacing) + "^".PadLeft(spacing));
done = true;
break;
}
}
}
if (done) {
break;
}
onePositions.Add(i);
}
}
Console.ReadKey();
}
static String getByteString(byte b) {
var s = new char[8];
for (var i=0; i<s.Length; i++) {
s[i] = ((b & _bits[i]) > 0 ? '1' : '0');
}
return new String(s);
}
static bool isOne(byte[] bytes, int i)
{
var byteIndex = i / 8;
var bitIndex = i % 8;
return (bytes[byteIndex] & _bits[bitIndex]) > 0;
}
}
}
I thought I'd add one comment before posting the 22nd naive solution to the problem. For the naive solution, we don't need to show that the number of 1's in the string is at most O(log(n)), but rather that it is at most O(sqrt(n*log(n)).
Solver:
def solve(Str):
indexes=[]
#O(n) setup
for i in range(len(Str)):
if Str[i]=='1':
indexes.append(i)
#O((number of 1's)^2) processing
for i in range(len(indexes)):
for j in range(i+1, len(indexes)):
indexDiff = indexes[j] - indexes[i]
k=indexes[j] + indexDiff
if k<len(Str) and Str[k]=='1':
return True
return False
It's basically a fair bit similar to flybywire's idea and implementation, though looking ahead instead of back.
Greedy String Builder:
#assumes final char hasn't been added, and would be a 1
def lastCharMakesSolvable(Str):
endIndex=len(Str)
j=endIndex-1
while j-(endIndex-j) >= 0:
k=j-(endIndex-j)
if k >= 0 and Str[k]=='1' and Str[j]=='1':
return True
j=j-1
return False
def expandString(StartString=''):
if lastCharMakesSolvable(StartString):
return StartString + '0'
return StartString + '1'
n=1
BaseStr=""
lastCount=0
while n<1000000:
BaseStr=expandString(BaseStr)
count=BaseStr.count('1')
if count != lastCount:
print(len(BaseStr), count)
lastCount=count
n=n+1
(In my defense, I'm still in the 'learn python' stage of understanding)
Also, potentially useful output from the greedy building of strings, there's a rather consistent jump after hitting a power of 2 in the number of 1's... which I was not willing to wait around to witness hitting 2096.
strlength # of 1's
1 1
2 2
4 3
5 4
10 5
14 8
28 9
41 16
82 17
122 32
244 33
365 64
730 65
1094 128
2188 129
3281 256
6562 257
9842 512
19684 513
29525 1024
I'll try to present a mathematical approach. This is more a beginning than an end, so any help, comment, or even contradiction - will be deeply appreciated. However, if this approach is proven - the algorithm is a straight-forward search in the string.
Given a fixed number of spaces k and a string S, the search for a k-spaced-triplet takes O(n) - We simply test for every 0<=i<=(n-2k) if S[i]==S[i+k]==S[i+2k]. The test takes O(1) and we do it n-k times where k is a constant, so it takes O(n-k)=O(n).
Let us assume that there is an Inverse Proportion between the number of 1's and the maximum spaces we need to search for. That is, If there are many 1's, there must be a triplet and it must be quite dense; If there are only few 1's, The triplet (if any) can be quite sparse. In other words, I can prove that if I have enough 1's, such triplet must exist - and the more 1's I have, a more dense triplet must be found. This can be explained by the Pigeonhole principle - Hope to elaborate on this later.
Say have an upper bound k on the possible number of spaces I have to look for. Now, for each 1 located in S[i] we need to check for 1 in S[i-1] and S[i+1], S[i-2] and S[i+2], ... S[i-k] and S[i+k]. This takes O((k^2-k)/2)=O(k^2) for each 1 in S - due to Gauss' Series Summation Formula. Note that this differs from section 1 - I'm having k as an upper bound for the number of spaces, not as a constant space.
We need to prove O(n*log(n)). That is, we need to show that k*(number of 1's) is proportional to log(n).
If we can do that, the algorithm is trivial - for each 1 in S whose index is i, simply look for 1's from each side up to distance k. If two were found in the same distance, return i and k. Again, the tricky part would be finding k and proving the correctness.
I would really appreciate your comments here - I have been trying to find the relation between k and the number of 1's on my whiteboard, so far without success.
Assumption:
Just wrong, talking about log(n) number of upper limit of ones
EDIT:
Now I found that using Cantor numbers (if correct), density on set is (2/3)^Log_3(n) (what a weird function) and I agree, log(n)/n density is to strong.
If this is upper limit, there is algorhitm who solves this problem in at least O(n*(3/2)^(log(n)/log(3))) time complexity and O((3/2)^(log(n)/log(3))) space complexity. (check Justice's answer for algorhitm)
This is still by far better than O(n^2)
This function ((3/2)^(log(n)/log(3))) really looks like n*log(n) on first sight.
How did I get this formula?
Applaying Cantors number on string.
Supose that length of string is 3^p == n
At each step in generation of Cantor string you keep 2/3 of prevous number of ones. Apply this p times.
That mean (n * ((2/3)^p)) -> (((3^p)) * ((2/3)^p)) remaining ones and after simplification 2^p.
This mean 2^p ones in 3^p string -> (3/2)^p ones . Substitute p=log(n)/log(3) and get
((3/2)^(log(n)/log(3)))
How about a simple O(n) solution, with O(n^2) space? (Uses the assumption that all bitwise operators work in O(1).)
The algorithm basically works in four stages:
Stage 1: For each bit in your original number, find out how far away the ones are, but consider only one direction. (I considered all the bits in the direction of the least significant bit.)
Stage 2: Reverse the order of the bits in the input;
Stage 3: Re-run step 1 on the reversed input.
Stage 4: Compare the results from Stage 1 and Stage 3. If any bits are equally spaced above AND below we must have a hit.
Keep in mind that no step in the above algorithm takes longer than O(n). ^_^
As an added benefit, this algorithm will find ALL equally spaced ones from EVERY number. So for example if you get a result of "0x0005" then there are equally spaced ones at BOTH 1 and 3 units away
I didn't really try optimizing the code below, but it is compilable C# code that seems to work.
using System;
namespace ThreeNumbers
{
class Program
{
const int uint32Length = 32;
static void Main(string[] args)
{
Console.Write("Please enter your integer: ");
uint input = UInt32.Parse(Console.ReadLine());
uint[] distancesLower = Distances(input);
uint[] distancesHigher = Distances(Reverse(input));
PrintHits(input, distancesLower, distancesHigher);
}
/// <summary>
/// Returns an array showing how far the ones away from each bit in the input. Only
/// considers ones at lower signifcant bits. Index 0 represents the least significant bit
/// in the input. Index 1 represents the second least significant bit in the input and so
/// on. If a one is 3 away from the bit in question, then the third least significant bit
/// of the value will be sit.
///
/// As programed this algorithm needs: O(n) time, and O(n*log(n)) space.
/// (Where n is the number of bits in the input.)
/// </summary>
public static uint[] Distances(uint input)
{
uint[] distanceToOnes = new uint[uint32Length];
uint result = 0;
//Sets how far each bit is from other ones. Going in the direction of LSB to MSB
for (uint bitIndex = 1, arrayIndex = 0; bitIndex != 0; bitIndex <<= 1, ++arrayIndex)
{
distanceToOnes[arrayIndex] = result;
result <<= 1;
if ((input & bitIndex) != 0)
{
result |= 1;
}
}
return distanceToOnes;
}
/// <summary>
/// Reverses the bits in the input.
///
/// As programmed this algorithm needs O(n) time and O(n) space.
/// (Where n is the number of bits in the input.)
/// </summary>
/// <param name="input"></param>
/// <returns></returns>
public static uint Reverse(uint input)
{
uint reversedInput = 0;
for (uint bitIndex = 1; bitIndex != 0; bitIndex <<= 1)
{
reversedInput <<= 1;
reversedInput |= (uint)((input & bitIndex) != 0 ? 1 : 0);
}
return reversedInput;
}
/// <summary>
/// Goes through each bit in the input, to check if there are any bits equally far away in
/// the distancesLower and distancesHigher
/// </summary>
public static void PrintHits(uint input, uint[] distancesLower, uint[] distancesHigher)
{
const int offset = uint32Length - 1;
for (uint bitIndex = 1, arrayIndex = 0; bitIndex != 0; bitIndex <<= 1, ++arrayIndex)
{
//hits checks if any bits are equally spaced away from our current value
bool isBitSet = (input & bitIndex) != 0;
uint hits = distancesLower[arrayIndex] & distancesHigher[offset - arrayIndex];
if (isBitSet && (hits != 0))
{
Console.WriteLine(String.Format("The {0}-th LSB has hits 0x{1:x4} away", arrayIndex + 1, hits));
}
}
}
}
}
Someone will probably comment that for any sufficiently large number, bitwise operations cannot be done in O(1). You'd be right. However, I'd conjecture that every solution that uses addition, subtraction, multiplication, or division (which cannot be done by shifting) would also have that problem.
Below is a solution. There could be some little mistakes here and there, but the idea is sound.
Edit: It's not n * log(n)
PSEUDO CODE:
foreach character in the string
if the character equals 1 {
if length cache > 0 { //we can skip the first one
foreach location in the cache { //last in first out kind of order
if ((currentlocation + (currentlocation - location)) < length string)
if (string[(currentlocation + (currentlocation - location))] equals 1)
return found evenly spaced string
else
break;
}
}
remember the location of this character in a some sort of cache.
}
return didn't find evenly spaced string
C# code:
public static Boolean FindThreeEvenlySpacedOnes(String str) {
List<int> cache = new List<int>();
for (var x = 0; x < str.Length; x++) {
if (str[x] == '1') {
if (cache.Count > 0) {
for (var i = cache.Count - 1; i > 0; i--) {
if ((x + (x - cache[i])) >= str.Length)
break;
if (str[(x + (x - cache[i]))] == '1')
return true;
}
}
cache.Add(x);
}
}
return false;
}
How it works:
iteration 1:
x
|
101101001
// the location of this 1 is stored in the cache
iteration 2:
x
|
101101001
iteration 3:
a x b
| | |
101101001
//we retrieve location a out of the cache and then based on a
//we calculate b and check if te string contains a 1 on location b
//and of course we store x in the cache because it's a 1
iteration 4:
axb
|||
101101001
a x b
| | |
101101001
iteration 5:
x
|
101101001
iteration 6:
a x b
| | |
101101001
a x b
| | |
101101001
//return found evenly spaced string
Obviously we need to at least check bunches of triplets at the same time, so we need to compress the checks somehow. I have a candidate algorithm, but analyzing the time complexity is beyond my ability*time threshold.
Build a tree where each node has three children and each node contains the total number of 1's at its leaves. Build a linked list over the 1's, as well. Assign each node an allowed cost proportional to the range it covers. As long as the time we spend at each node is within budget, we'll have an O(n lg n) algorithm.
--
Start at the root. If the square of the total number of 1's below it is less than its allowed cost, apply the naive algorithm. Otherwise recurse on its children.
Now we have either returned within budget, or we know that there are no valid triplets entirely contained within one of the children. Therefore we must check the inter-node triplets.
Now things get incredibly messy. We essentially want to recurse on the potential sets of children while limiting the range. As soon as the range is constrained enough that the naive algorithm will run under budget, you do it. Enjoy implementing this, because I guarantee it will be tedious. There's like a dozen cases.
--
The reason I think that algorithm will work is because the sequences without valid triplets appear to go alternate between bunches of 1's and lots of 0's. It effectively splits the nearby search space, and the tree emulates that splitting.
The run time of the algorithm is not obvious, at all. It relies on the non-trivial properties of the sequence. If the 1's are really sparse then the naive algorithm will work under budget. If the 1's are dense, then a match should be found right away. But if the density is 'just right' (eg. near ~n^0.63, which you can achieve by setting all bits at positions with no '2' digit in base 3), I don't know if it will work. You would have to prove that the splitting effect is strong enough.
No theoretical answer here, but I wrote a quick Java program to explore the running-time behavior as a function of k and n, where n is the total bit length and k is the number of 1's. I'm with a few of the answerers who are saying that the "regular" algorithm that checks all the pairs of bit positions and looks for the 3rd bit, even though it would require O(k^2) in the worst case, in reality because the worst-case needs sparse bitstrings, is O(n ln n).
Anyway here's the program, below. It's a Monte-Carlo style program which runs a large number of trials NTRIALS for constant n, and randomly generates bitsets for a range of k-values using Bernoulli processes with ones-density constrained between limits that can be specified, and records the running time of finding or failing to find a triplet of evenly spaced ones, time measured in steps NOT in CPU time. I ran it for n=64, 256, 1024, 4096, 16384* (still running), first a test run with 500000 trials to see which k-values take the longest running time, then another test with 5000000 trials with narrowed ones-density focus to see what those values look like. The longest running times do happen with very sparse density (e.g. for n=4096 the running time peaks are in the k=16-64 range, with a gentle peak for mean runtime at 4212 steps # k=31, max runtime peaked at 5101 steps # k=58). It looks like it would take extremely large values of N for the worst-case O(k^2) step to become larger than the O(n) step where you scan the bitstring to find the 1's position indices.
package com.example.math;
import java.io.PrintStream;
import java.util.BitSet;
import java.util.Random;
public class EvenlySpacedOnesTest {
static public class StatisticalSummary
{
private int n=0;
private double min=Double.POSITIVE_INFINITY;
private double max=Double.NEGATIVE_INFINITY;
private double mean=0;
private double S=0;
public StatisticalSummary() {}
public void add(double x) {
min = Math.min(min, x);
max = Math.max(max, x);
++n;
double newMean = mean + (x-mean)/n;
S += (x-newMean)*(x-mean);
// this algorithm for mean,std dev based on Knuth TAOCP vol 2
mean = newMean;
}
public double getMax() { return (n>0)?max:Double.NaN; }
public double getMin() { return (n>0)?min:Double.NaN; }
public int getCount() { return n; }
public double getMean() { return (n>0)?mean:Double.NaN; }
public double getStdDev() { return (n>0)?Math.sqrt(S/n):Double.NaN; }
// some may quibble and use n-1 for sample std dev vs population std dev
public static void printOut(PrintStream ps, StatisticalSummary[] statistics) {
for (int i = 0; i < statistics.length; ++i)
{
StatisticalSummary summary = statistics[i];
ps.printf("%d\t%d\t%.0f\t%.0f\t%.5f\t%.5f\n",
i,
summary.getCount(),
summary.getMin(),
summary.getMax(),
summary.getMean(),
summary.getStdDev());
}
}
}
public interface RandomBernoulliProcess // see http://en.wikipedia.org/wiki/Bernoulli_process
{
public void setProbability(double d);
public boolean getNextBoolean();
}
static public class Bernoulli implements RandomBernoulliProcess
{
final private Random r = new Random();
private double p = 0.5;
public boolean getNextBoolean() { return r.nextDouble() < p; }
public void setProbability(double d) { p = d; }
}
static public class TestResult {
final public int k;
final public int nsteps;
public TestResult(int k, int nsteps) { this.k=k; this.nsteps=nsteps; }
}
////////////
final private int n;
final private int ntrials;
final private double pmin;
final private double pmax;
final private Random random = new Random();
final private Bernoulli bernoulli = new Bernoulli();
final private BitSet bits;
public EvenlySpacedOnesTest(int n, int ntrials, double pmin, double pmax) {
this.n=n; this.ntrials=ntrials; this.pmin=pmin; this.pmax=pmax;
this.bits = new BitSet(n);
}
/*
* generate random bit string
*/
private int generateBits()
{
int k = 0; // # of 1's
for (int i = 0; i < n; ++i)
{
boolean b = bernoulli.getNextBoolean();
this.bits.set(i, b);
if (b) ++k;
}
return k;
}
private int findEvenlySpacedOnes(int k, int[] pos)
{
int[] bitPosition = new int[k];
for (int i = 0, j = 0; i < n; ++i)
{
if (this.bits.get(i))
{
bitPosition[j++] = i;
}
}
int nsteps = n; // first, it takes N operations to find the bit positions.
boolean found = false;
if (k >= 3) // don't bother doing anything if there are less than 3 ones. :(
{
int lastBitSetPosition = bitPosition[k-1];
for (int j1 = 0; !found && j1 < k; ++j1)
{
pos[0] = bitPosition[j1];
for (int j2 = j1+1; !found && j2 < k; ++j2)
{
pos[1] = bitPosition[j2];
++nsteps;
pos[2] = 2*pos[1]-pos[0];
// calculate 3rd bit index that might be set;
// the other two indices point to bits that are set
if (pos[2] > lastBitSetPosition)
break;
// loop inner loop until we go out of bounds
found = this.bits.get(pos[2]);
// we're done if we find a third 1!
}
}
}
if (!found)
pos[0]=-1;
return nsteps;
}
/*
* run an algorithm that finds evenly spaced ones and returns # of steps.
*/
public TestResult run()
{
bernoulli.setProbability(pmin + (pmax-pmin)*random.nextDouble());
// probability of bernoulli process is randomly distributed between pmin and pmax
// generate bit string.
int k = generateBits();
int[] pos = new int[3];
int nsteps = findEvenlySpacedOnes(k, pos);
return new TestResult(k, nsteps);
}
public static void main(String[] args)
{
int n;
int ntrials;
double pmin = 0, pmax = 1;
try {
n = Integer.parseInt(args[0]);
ntrials = Integer.parseInt(args[1]);
if (args.length >= 3)
pmin = Double.parseDouble(args[2]);
if (args.length >= 4)
pmax = Double.parseDouble(args[3]);
}
catch (Exception e)
{
System.out.println("usage: EvenlySpacedOnesTest N NTRIALS [pmin [pmax]]");
System.exit(0);
return; // make the compiler happy
}
final StatisticalSummary[] statistics;
statistics=new StatisticalSummary[n+1];
for (int i = 0; i <= n; ++i)
{
statistics[i] = new StatisticalSummary();
}
EvenlySpacedOnesTest test = new EvenlySpacedOnesTest(n, ntrials, pmin, pmax);
int printInterval=100000;
int nextPrint = printInterval;
for (int i = 0; i < ntrials; ++i)
{
TestResult result = test.run();
statistics[result.k].add(result.nsteps);
if (i == nextPrint)
{
System.err.println(i);
nextPrint += printInterval;
}
}
StatisticalSummary.printOut(System.out, statistics);
}
}
# <algorithm>
def contains_evenly_spaced?(input)
return false if input.size < 3
one_indices = []
input.each_with_index do |digit, index|
next if digit == 0
one_indices << index
end
return false if one_indices.size < 3
previous_indexes = []
one_indices.each do |index|
if !previous_indexes.empty?
previous_indexes.each do |previous_index|
multiple = index - previous_index
success_index = index + multiple
return true if input[success_index] == 1
end
end
previous_indexes << index
end
return false
end
# </algorithm>
def parse_input(input)
input.chars.map { |c| c.to_i }
end
I'm having trouble with the worst-case scenarios with millions of digits. Fuzzing from /dev/urandom essentially gives you O(n), but I know the worst case is worse than that. I just can't tell how much worse. For small n, it's trivial to find inputs at around 3*n*log(n), but it's surprisingly hard to differentiate those from some other order of growth for this particular problem.
Can anyone who was working on worst-case inputs generate a string with length greater than say, one hundred thousand?
An adaptation of the Rabin-Karp algorithm could be possible for you.
Its complexity is 0(n) so it could help you.
Take a look http://en.wikipedia.org/wiki/Rabin-Karp_string_search_algorithm
Could this be a solution? I', not sure if it's O(nlogn) but in my opinion it's better than O(n²) because the the only way not to find a triple would be a prime number distribution.
There's room for improvement, the second found 1 could be the next first 1. Also no error checking.
#include <iostream>
#include <string>
int findIt(std::string toCheck) {
for (int i=0; i<toCheck.length(); i++) {
if (toCheck[i]=='1') {
std::cout << i << ": " << toCheck[i];
for (int j = i+1; j<toCheck.length(); j++) {
if (toCheck[j]=='1' && toCheck[(i+2*(j-i))] == '1') {
std::cout << ", " << j << ":" << toCheck[j] << ", " << (i+2*(j-i)) << ":" << toCheck[(i+2*(j-i))] << " found" << std::endl;
return 0;
}
}
}
}
return -1;
}
int main (int agrc, char* args[]) {
std::string toCheck("1001011");
findIt(toCheck);
std::cin.get();
return 0;
}
I think this algorithm has O(n log n) complexity (C++, DevStudio 2k5). Now, I don't know the details of how to analyse an algorithm to determine its complexity, so I have added some metric gathering information to the code. The code counts the number of tests done on the sequence of 1's and 0's for any given input (hopefully, I've not made a balls of the algorithm). We can compare the actual number of tests against the O value and see if there's a correlation.
#include <iostream>
using namespace std;
bool HasEvenBits (string &sequence, int &num_compares)
{
bool
has_even_bits = false;
num_compares = 0;
for (unsigned i = 1 ; i <= (sequence.length () - 1) / 2 ; ++i)
{
for (unsigned j = 0 ; j < sequence.length () - 2 * i ; ++j)
{
++num_compares;
if (sequence [j] == '1' && sequence [j + i] == '1' && sequence [j + i * 2] == '1')
{
has_even_bits = true;
// we could 'break' here, but I want to know the worst case scenario so keep going to the end
}
}
}
return has_even_bits;
}
int main ()
{
int
count;
string
input = "111";
for (int i = 3 ; i < 32 ; ++i)
{
HasEvenBits (input, count);
cout << i << ", " << count << endl;
input += "0";
}
}
This program outputs the number of tests for each string length up to 32 characters. Here's the results:
n Tests n log (n)
=====================
3 1 1.43
4 2 2.41
5 4 3.49
6 6 4.67
7 9 5.92
8 12 7.22
9 16 8.59
10 20 10.00
11 25 11.46
12 30 12.95
13 36 14.48
14 42 16.05
15 49 17.64
16 56 19.27
17 64 20.92
18 72 22.59
19 81 24.30
20 90 26.02
21 100 27.77
22 110 29.53
23 121 31.32
24 132 33.13
25 144 34.95
26 156 36.79
27 169 38.65
28 182 40.52
29 196 42.41
30 210 44.31
31 225 46.23
I've added the 'n log n' values as well. Plot these using your graphing tool of choice to see a correlation between the two results. Does this analysis extend to all values of n? I don't know.