Related
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.
I need to generate binary numbers with the same quantity of ones (or zeros) in random order.
Does anyone know any efficient algorithm for fixed-length binary numbers?
Example for 2 ones and 4 digits (just to be more clear):
1100
1010
1001
0110
0101
0011
UPDATE
Random order w/o repetitions is significant. Sequence of binary numbers required, not single permutation.
If you have enough memory to store all the possible bit sequences, and you don't mind generating them all before you have the first result, then the solution would be to use some efficient generator to produce all possible sequences into a vector and then shuffle the vector using the Fisher-Yates shuffle. That's easy and unbiased (as long as you use a good random number generator to do the shuffle) but it can use a lot of memory if n is large, particularly if you are not sure you will need to complete the iteration.
But there are a couple of solutions which do not require keeping all the possible words in memory. (C implementations of the two solutions follow the text.)
1. Bit shuffle an enumeration
The fastest one (I think) is to first generate a random shuffle of bit values, and then iterate over the possible words one at a time applying the shuffle to the bits of each value. In order to avoid the complication of shuffling actual bits, the words can be generated in a Gray code order in which only two bit positions are changed from one word to the next. (This is also known as a "revolving-door" iteration because as each new 1 is added, some other 1 must be removed.) This allows the bit mask to be updated rapidly, but it means that successive entries are highly correlated, which may be unsuitable for some purposes. Also, for small values of n the number of possible bit shuffles is very limited, so there will not be a lot of different sequences produced. (For example, for the case where n is 4 and k is 2, there are 6 possible words which could be sequenced in 6! (720) different ways, but there are only 4! (24) bit-shuffles. This could be ameliorated slightly by starting the iteration at a random position in the sequence.)
It is always possible to find a Gray code. Here's an example for n=6, k=3: (The bold bits are swapped at each step. I wanted to underline them but for some inexplicable reason SO allows strikethrough but not underline.)
111000 010110 100011 010101
101100 001110 010011 001101
011100 101010 001011 101001
110100 011010 000111 011001
100110 110010 100101 110001
This sequence can be produced by a recursive algorithm similar to that suggested by #JasonBoubin -- the only difference is that the second half of each recursion needs to be produced in reverse order -- but it's convenient to use a non-recursive version of the algorithm. The one in the sample code below comes from Frank Ruskey's unpublished manuscript on Combinatorial Generation (Algorithm 5.7 on page 130). I modified it to use 0-based indexing, as well as adding the code to keep track of the binary representations.
2. Randomly generate an integer sequence and convert it to combinations
The "more" random but somewhat slower solution is to produce a shuffled list of enumeration indices (which are sequential integers in [0, n choose k)) and then find the word corresponding to each index.
The simplest pseudo-random way to produce a shuffled list of integers in a contiguous range is to use a randomly-chosen Linear Congruential Generator (LCG). An LCG is the recursive sequence xi = (a * xi-1 + c) mod m. If m is a power of 2, a mod 4 is 1 and c mod 2 is 1, then that recursion will cycle through all 2m possible values. To cycle through the range [0, n choose k), we simply select m to be the next larger power of 2, and then skip any values which are not in the desired range. (That will be fewer than half the values produced, for obvious reasons.)
To convert the enumeration index into an actual word, we perform a binomial decomposition of the index based on the fact that the set of n choose k words consists of n-1 choose k words starting with a 0 and n-1 choose k-1 words starting with a 1. So to produce the ith word:
if i < n-1 choose k we output a 0 and then the ith word in the set of n-1 bit words with k bits set;
otherwise, we output a 1 and then subtract n-1 choose k from i as the index into the set of n-1 bit words with k-1 bits set.
It's convenient to precompute all the useful binomial coefficients.
LCGs suffer from the disadvantage that they are quite easy to predict after the first few terms are seen. Also, some of the randomly-selected values of a and c will produce index sequences where successive indices are highly correlated. (Also, the low-order bits are always quite non-random.) Some of these problems could be slightly ameliorated by also applying a random bit-shuffle to the final result. This is not illustrated in the code below but it would slow things down very little and it should be obvious how to do it. (It basically consists of replacing 1UL<<n with a table lookup into the shuffled bits).
The C code below uses some optimizations which make it a bit challenging to read. The binomial coefficients are stored in a lower-diagonal array:
row
index
[ 0] 1
[ 1] 1 1
[ 3] 1 2 1
[ 6] 1 3 3 1
[10] 1 4 6 4 1
As can be seen, the array index for binom(n, k) is n(n+1)/2 + k, and if we have that index, we can find binom(n-1, k) by simply subtracting n, and binom(n-1, k-1) by subtracting n+1. In order to avoid needing to store zeros in the array, we make sure that we never look up a binomial coefficient where k is negative or greater than n. In particular, if we have arrived at a point in the recursion where k == n or k == 0, we can definitely know that the index to look up is 0, because there is only one possible word. Furthermore, index 0 in the set of words with some n and k
will consist precisely of n-k zeros followed by k ones, which is the n-bit binary representation of 2k-1. By short-cutting the algorithm when the index reaches 0, we can avoid having to worry about the cases where one of binom(n-1, k) or binom(n-1, k-1) is not a valid index.
C code for the two solutions
Gray code with shuffled bits
void gray_combs(int n, int k) {
/* bit[i] is the ith shuffled bit */
uint32_t bit[n+1];
{
uint32_t mask = 1;
for (int i = 0; i < n; ++i, mask <<= 1)
bit[i] = mask;
bit[n] = 0;
shuffle(bit, n);
}
/* comb[i] for 0 <= i < k is the index of the ith bit
* in the current combination. comb[k] is a sentinel. */
int comb[k + 1];
for (int i = 0; i < k; ++i) comb[i] = i;
comb[k] = n;
/* Initial word has the first k (shuffled) bits set */
uint32_t word = 0;
for (int i = 0; i < k; ++i) word |= bit[i];
/* Now iterate over all combinations */
int j = k - 1; /* See Ruskey for meaning of j */
do {
handle(word, n);
if (j < 0) {
word ^= bit[comb[0]] | bit[comb[0] - 1];
if (--comb[0] == 0) j += 2;
}
else if (comb[j + 1] == comb[j] + 1) {
word ^= bit[comb[j + 1]] | bit[j];
comb[j + 1] = comb[j]; comb[j] = j;
if (comb[j + 1] == comb[j] + 1) j += 2;
}
else if (j > 0) {
word ^= bit[comb[j - 1]] | bit[comb[j] + 1];
comb[j - 1] = comb[j]; ++comb[j];
j -= 2;
}
else {
word ^= bit[comb[j]] | bit[comb[j] + 1];
++comb[j];
}
} while (comb[k] == n);
}
LCG with enumeration index to word conversion
static const uint32_t* binom(unsigned n, unsigned k) {
static const uint32_t b[] = {
1,
1, 1,
1, 2, 1,
1, 3, 3, 1,
1, 4, 6, 4, 1,
1, 5, 10, 10, 5, 1,
1, 6, 15, 20, 15, 6, 1,
// ... elided for space
};
return &b[n * (n + 1) / 2 + k];
}
static uint32_t enumerate(const uint32_t* b, uint32_t r, unsigned n, unsigned k) {
uint32_t rv = 0;
while (r) {
do {
b -= n;
--n;
} while (r < *b);
r -= *b;
--b;
--k;
rv |= 1UL << n;
}
return rv + (1UL << k) - 1;
}
static bool lcg_combs(unsigned n, unsigned k) {
const uint32_t* b = binom(n, k);
uint32_t count = *b;
uint32_t m = 1; while (m < count) m <<= 1;
uint32_t a = 4 * randrange(1, m / 4) + 1;
uint32_t c = 2 * randrange(0, m / 2) + 1;
uint32_t x = randrange(0, m);
while (count--) {
do
x = (a * x + c) & (m - 1);
while (x >= *b);
handle(enumerate(b, x, n, k), n);
}
return true;
}
Note: I didn't include the implementation of randrange or shuffle; code is readily available. randrange(low, lim) produces a random integer in the range [low, lim); shuffle(vec, n) randomly shuffles the integer vector vecof length n.
Also, the the loop calls handle(word, n) for each generated word. That must must be replaced with whatever is to be done with each combination.
With handle defined as a function which does nothing, gray_combs took 150 milliseconds on my laptop to find all 40,116,600 28-bit words with 14 bits set. lcg_combs took 5.5 seconds.
Integers with exactly k bits set are easy to generate in order.
You can do that, and then change the order by applying a bit-permutation to the results (see below), for example here's a randomly generated 16-bit (you should pick one with the right number of bits, based on the word size not on the number of set bits) bit-permutation (not tested):
uint permute(uint x) {
x = bit_permute_step(x, 0x00005110, 1); // Butterfly, stage 0
x = bit_permute_step(x, 0x00000709, 4); // Butterfly, stage 2
x = bit_permute_step(x, 0x000000a1, 8); // Butterfly, stage 3
x = bit_permute_step(x, 0x00005404, 1); // Butterfly, stage 0
x = bit_permute_step(x, 0x00000231, 2); // Butterfly, stage 1
return x;
}
uint bit_permute_step(uint x, uint m, int shift) {
uint t;
t = ((x >> shift) ^ x) & m;
x = (x ^ t) ^ (t << shift);
return x;
}
Generating the re-ordered sequence is easy:
uint i = (1u << k) - 1;
uint max = i << (wordsize - k);
do
{
yield permute(i);
i = nextPermutation(i);
} while (i != max);
yield permute(i); // for max
Where nextPermutation comes from the linked question,
uint nextPermutation(uint v) {
uint t = (v | (v - 1)) + 1;
uint w = t | ((((t & -t) / (v & -v)) >> 1) - 1);
return w;
}
The bit-permutation should be chosen as a random permutation (eg take 0..(wordsize-1) and shuffle) and then converted to bfly masks (I used programming.sirrida.de/calcperm.php), not as randomly generated bfly masks.
I think you can use Heap's algorithm. This algorithm generates all possible permutations of n objects. Just create simple array and use algorithm for generating all possible permutations.
This algorithm is non effective if you want to iterate over binary numbers with BINARY operations. For binary operations you can use LFSR.
LFSR is a simple method for iteration over all numbers. I think you can do some simple modifications for generations fixed size zeros numbers with LFSR.
How about this solution in Python which does permutations?
from itertools import permutations
fixed_length = 4
perms = [''.join(p) for p in permutations('11' + '0' * (fixed_length - 2))]
unique_perms = set(perms)
This would return the numbers as strings, easily convertible with int(num, 2).
As for efficiency, running this took 0.021 milliseconds on my machine.
You can modify the general permutation algorithm to work with binary. Here's an implementation in C++:
#include<iostream>
#include<string>
#include<iostream>
void binaryPermutation(int ones, int digits, std::string current){
if(digits <= 0 && ones <= 0){
std::cout<<current<<std::endl;
}
else if(digits > 0){
if(ones > 0){
binaryPermutation(ones-1, digits-1, current+"1");
}
binaryPermutation(ones, digits-1, current+"0");
}
}
int main()
{
binaryPermutation(2, 4, "");
return 0;
}
This code outputs the following:
1100
1010
1001
0110
0101
0011
You can modify it to store these outputs in a collection or do something other than simply print them.
Actually, this question can be generalized as below:
Find all possible combinations from a given set of elements, which meets
a certain criteria.
So, any good algorithms?
There are only 16 possibilities (and one of those is to add together "none of them", which ain't gonna give you 24), so the old-fashioned "brute force" algorithm looks pretty good to me:
for (unsigned int choice = 1; choice < 16; ++choice) {
int sum = 0;
if (choice & 1) sum += elements[0];
if (choice & 2) sum += elements[1];
if (choice & 4) sum += elements[2];
if (choice & 8) sum += elements[3];
if (sum == 24) {
// we have a winner
}
}
In the completely general form of your problem, the only way to tell whether a combination meets "certain criteria" is to evaluate those criteria for every single combination. Given more information about the criteria, maybe you could work out some ways to avoid testing every combination and build an algorithm accordingly, but not without those details. So again, brute force is king.
There are two interesting explanations about the sum problem, both in Wikipedia and MathWorld.
In the case of the first question you asked, the first answer is good for a limited number of elements. You should realize that the reason Mr. Jessop used 16 as the boundary for his loop is because this is 2^4, where 4 is the number of elements in your set. If you had 100 elements, the loop limit would become 2^100 and your algorithm would literally take forever to finish.
In the case of a bounded sum, you should consider a depth first search, because when the sum of elements exceeds the sum you are looking for, you can prune your branch and backtrack.
In the case of the generic question, finding the subset of elements that satisfy certain criteria, this is known as the Knapsack problem, which is known to be NP-Complete. Given that, there is no algorithm that will solve it in less than exponential time.
Nevertheless, there are several heuristics that bring good results to the table, including (but not limited to) genetic algorithms (one I personally like, for I wrote a book on them) and dynamic programming. A simple search in Google will show many scientific papers that describe different solutions for this problem.
Find all possible combinations from a given set of elements, which
meets a certain criteria
If i understood you right, this code will helpful for you:
>>> from itertools import combinations as combi
>>> combi.__doc__
'combinations(iterable, r) --> combinations object\n\nReturn successive r-length
combinations of elements in the iterable.\n\ncombinations(range(4), 3) --> (0,1
,2), (0,1,3), (0,2,3), (1,2,3)'
>>> set = range(4)
>>> set
[0, 1, 2, 3]
>>> criteria = range(3)
>>> criteria
[0, 1, 2]
>>> for tuple in list(combi(set, len(criteria))):
... if cmp(list(tuple), criteria) == 0:
... print 'criteria exists in tuple: ', tuple
...
criteria exists in tuple: (0, 1, 2)
>>> list(combi(set, len(criteria)))
[(0, 1, 2), (0, 1, 3), (0, 2, 3), (1, 2, 3)]
Generally for a problem as this you have to try all posebilities, the thing you should do have the code abort the building of combiantion if you know it will not satesfie the criteria (if you criteria is that you do not have more then two blue balls, then you have to abort calculation that has more then two). Backtracing
def perm(set,permutation):
if lenght(set) == lenght(permutation):
print permutation
else:
for element in set:
if permutation.add(element) == criteria:
perm(sett,permutation)
else:
permutation.pop() //remove the element added in the if
The set of input numbers matters, as you can tell as soon as you allow e.g. negative numbers, imaginary numbers, rational numbers etc in your start set. You could also restrict to e.g. all even numbers, all odd number inputs etc.
That means that it's hard to build something deductive. You need brute force, a.k.a. try every combination etc.
In this particular problem you could build an algoritm that recurses - e.g. find every combination of 3 Int ( 1,22) that add up to 23, then add 1, every combination that add to 22 and add 2 etc. Which can again be broken into every combination of 2 that add up to 21 etc. You need to decide if you can count same number twice.
Once you have that you have a recursive function to call -
combinations( 24 , 4 ) = combinations( 23, 3 ) + combinations( 22, 3 ) + ... combinations( 4, 3 );
combinations( 23 , 3 ) = combinations( 22, 2 ) + ... combinations( 3, 2 );
etc
This works well except you have to be careful around repeating numbers in the recursion.
private int[][] work()
{
const int target = 24;
List<int[]> combos = new List<int[]>();
for(int i = 0; i < 9; i++)
for(int x = 0; x < 9; x++)
for(int y = 0; y < 9; y++)
for (int z = 0; z < 9; z++)
{
int res = x + y + z + i;
if (res == target)
{
combos.Add(new int[] { x, y, z, i });
}
}
return combos.ToArray();
}
It works instantly, but there probably are better methods rather than 'guess and check'. All I am doing is looping through every possibility, adding them all together, and seeing if it comes out to the target value.
If i understand your question correctly, what you are asking for is called "Permutations" or the number (N) of possible ways to arrange (X) numbers taken from a set of (Y) numbers.
N = Y! / (Y - X)!
I don't know if this will help, but this is a solution I came up with for an assignment on permutations.
You have an input of : 123 (string) using the substr functions
1) put each number of the input into an array
array[N1,N2,N3,...]
2)Create a swap function
function swap(Number A, Number B)
{
temp = Number B
Number B = Number A
Number A = temp
}
3)This algorithm uses the swap function to move the numbers around until all permutations are done.
original_string= '123'
temp_string=''
While( temp_string != original_string)
{
swap(array element[i], array element[i+1])
if (i == 1)
i == 0
temp_string = array.toString
i++
}
Hopefully you can follow my pseudo code, but this works at least for 3 digit permutations
(n X n )
built up a square matrix of nxn
and print all together its corresponding crossed values
e.g.
1 2 3 4
1 11 12 13 14
2 .. .. .. ..
3 ..
4 .. ..
I was trying to solve a problem on SPOJ. We are required to calculate the nth twin prime pair( primes differing by 2). n can be as large as 10^5. I tried a precalculation using a sieve, I had to sieve up to 10^8 to get the maximum n twin prime, but the time limit is strict(2s) and it times out. I noticed people have solved it in 0.00 seconds, so i looked around for a formula on google, and couldnt get anything helpful. Could someone please guide me?
Thanks in advance!!
Out of curiosity, I solved the problem using two variants of a Sieve of Eratosthenes. The first variant completed on the testing machine in 0.93s and the second in 0.24s. For comparison, on my computer, the first finished in 0.08s and the second in 0.04s.
The first was a standard sieve on the odd numbers, the second a slightly more elaborate sieve omitting also the multiples of 3 in addition to the even numbers.
The testing machines of SPOJ are old and slow, so a programme runs much longer on them than on a typical recent box; and they have small caches, therefore it is important to keep the computation small.
Doing that, a Sieve of Eratosthenes is easily fast enough. However, it is really important to keep memory usage small. The first variant, using one byte per number, gave "Time limit exceeded" on SPOJ, but ran in 0.12s on my box. So, given the characteristics of the SPOJ testing machines, use a bit-sieve to solve it in the given time.
On the SPOJ machine, I got a significant speedup (running time 0.14s) by further reducing the space of the sieve by half. Since - except for the first pair (3,5) - all prime twins have the form (6*k-1, 6*k+1), and you need not know which of the two numbers is composite if k doesn't give rise to a twin prime pair, it is sufficient to sieve only the indices k.
(6*k + 1 is divisible by 5 if and only if k = 5*m + 4 for some m, and 6*k - 1 is divisible by 5 if and only if k = 5*m+1 for some m, so 5 would mark off 5*m ± 1, m >= 1 as not giving rise to twin primes. Similarly, 6*k+1 is divisible by 13 if and only if k = 13*m + 2 for some m and 6*k - 1 if and only if k = 13*m - 2 for some m, so 13 would mark off 13*m ± 2.)
This doesn't change the number of markings, so with a sufficiently large cache, the change in running time is small, but for small caches, it's a significant speedup.
One more thing, though. Your limit of 108 is way too high. I used a lower limit (20 million) that doesn't overestimate the 100,000th twin prime pair by so much. With a limit of 108, the first variant would certainly not have finished in time, the second probably not.
With the reduced limit, a Sieve of Atkin needs to be somewhat optimised to beat the Eratosthenes variant omitting even numbers and multiples of 3, a naive implementation will be significantly slower.
Some remarks concerning your (wikipedia's pseudocode) Atkin sieve:
#define limit 100000000
int prime1[MAXN];
int prime2[MAXN];
You don't need the second array, the larger partner of a prime twin pair can easily be computed from the smaller. You're wasting space and destroy cache locality reading from two arrays. (That's minor compared to the time needed for sieving, though.)
int root = ceil(sqrt(limit));
bool sieve[limit];
On many operating systems nowadays, that is an instant segfault, even with a reduced limit. The stack size is often limited to 8MB or less. Arrays of that size should be allocated on the heap.
As mentioned above, using one bool per number makes the programme run far slower than necessary. You should use a std::bitset or std::vector<bool> or twiddle the bits yourself. Also it is advisable to omit at least the even numbers.
for (int x = 1; x <= root; x++)
{
for (int y = 1; y <= root; y++)
{
//Main part of Sieve of Atkin
int n = (4*x*x)+(y*y);
if (n <= limit && (n % 12 == 1 || n % 12 == 5)) sieve[n] ^= true;
n = (3*x*x)+(y*y);
if (n <= limit && n % 12 == 7) sieve[n] ^= true;
n = (3*x*x)-(y*y);
if (x > y && n <= limit && n % 12 == 11) sieve[n] ^= true;
}
}
This is horribly inefficient. It tries far too many x-y-combinations, for each combination it does three or four divisions to check the remainder modulo 12 and it hops back and forth in the array.
Separate the different quadratics.
For 4*x^2 + y^2, it is evident that you need only consider x < sqrt(limit)/2 and odd y. Then the remainder modulo 12 is 1, 5, or 9. If the remainder is 9, then 4*x^2 + y^2 is actually a multiple of 9, so such a number would be eliminated as not square-free. However, it is preferable to omit the multiples of 3 from the sieve altogether and treat the cases n % 12 == 1 and n % 12 == 5 separately.
For 3*x^2 + y^2, it is evident that you need only consider x < sqrt(limit/3) and a little bit of thought reveals that x must be odd and y even (and not divisible by 3).
For 3*x^2 - y^2 with y < x, it is evident that you need only consider y < sqrt(limit/2). Looking at the remainders modulo 12, you see that y mustn't be divisible by 3 and x and y must have different parity.
I have got AC in 0.66s. As, there are solutions with 0.0s I assume better optimizations are possible, however, I describe my approach here.
I have used one basic optimization in Sieve of Eratosthenes. You know that 2 is the only even prime, using this you can reduce your computation time and memory for calculating primes by half.
Secondly, all the numbers which are twin primes will not be multiples of 2 and 3 (as they are primes!). So, those numbers will be of the form 6N+1 and 6N+5 (rest will not be primes for sure). 6N+5 = 6N+6-1 = 6(N+1)-1. So it can be seen that 6N+1 and 6N-1 can possibly be twin primes for N >= 1. So, you precompute all these values using the primes that you have calculated before. (Trivial case is 3 5)
Note: You don't need to calculate primes till 10^8, the upper limit is much lower.
[Edit: I can share my code if you want, but it would be better if you come up with a solution on your own. :)]
So basically, sieving up to 20,000,000 is enough, according to Wolfram Alpha. Use plain sieve of Eratosthenes, on odds, with vector<bool> in C++ (what language were you using BTW?).
Track the twin primes right inside the sieve loop. Store the lower prime of a pair in a separate vector as you find the twins, and if an out-of-order (smaller then previous) index is requested (and they are, contrary to the examples shown on the description page), just get the prime from this storage:
size_t n = 10000000, itop=2236;
vector<bool> s;
vector<int> twins;
s.resize(n, true);
int cnt, k1, k2, p1=3, p2, k=0;
cin >> cnt;
if( cnt-- > 0 )
{
cin >> k1;
for( size_t i=1; i < n; ++i ) // p=2i+1
{
if( s[i] )
{
p2 = 2*i+1;
if( p2-p1 == 2 ) { ++k; twins.push_back(p1); }
if( k==k1 )
{
cout << p1 << " " << p2 << endl;
......
etc. Got accept with 1.05 sec (0.18 sec on Ideone). Or untangle the logic - just pre-calculate 100,000 twin prime pairs right away, and access them in a separate loop afterwards (0.94 sec).
A description of an efficient algorithm to solve this can be found here # Programming Praxis entry Also, Scheme and Perl sample code are provided.
I precomputed a large list of primes using the Sieve of Eratosthenes, then iterated through the list counting items that were 2 less than their successor until finding n of them. Runs in 1.42 seconds at http://ideone.com/vYjuC. I too would like to know how to compute the answer in zero seconds.
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#define ISBITSET(x, i) (( x[i>>3] & (1<<(i&7)) ) != 0)
#define SETBIT(x, i) x[i>>3] |= (1<<(i&7));
#define CLEARBIT(x, i) x[i>>3] &= (1<<(i&7)) ^ 0xFF;
typedef struct list {
int data;
struct list *next;
} List;
List *insert(int data, List *next)
{
List *new;
new = malloc(sizeof(List));
new->data = data;
new->next = next;
return new;
}
List *reverse(List *list) {
List *new = NULL;
List *next;
while (list != NULL)
{
next = list->next;
list->next = new;
new = list;
list = next;
}
return new;
}
int length(List *xs)
{
int len = 0;
while (xs != NULL)
{
len += 1;
xs = xs->next;
}
return len;
}
List *primes(int n)
{
int m = (n-1) / 2;
char b[m/8+1];
int i = 0;
int p = 3;
List *ps = NULL;
int j;
ps = insert(2, ps);
memset(b, 255, sizeof(b));
while (p*p < n)
{
if (ISBITSET(b,i))
{
ps = insert(p, ps);
j = (p*p - 3) / 2;
while (j < m)
{
CLEARBIT(b, j);
j += p;
}
}
i += 1; p += 2;
}
while (i < m)
{
if (ISBITSET(b,i))
{
ps = insert(p, ps);
}
i += 1; p += 2;
}
return reverse(ps);
}
int nth_twin(int n, List *ps)
{
while (ps->next != NULL)
{
if (n == 0)
{
return ps->data - 1;
}
if (ps->next->data - ps->data == 2)
{
--n;
}
ps = ps->next;
}
return 0;
}
int main(int argc, char *argv[])
{
List *ps = primes(100000000);
printf("%d\n", nth_twin(100000, ps));
return 0;
}
this is what I have attempted. I have a string of TLEs.
bool mark [N];
vector <int> primeList;
void sieve ()
{
memset (mark, true, sizeof (mark));
mark [0] = mark [1] = false;
for ( int i = 4; i < N; i += 2 )
mark [i] = false;
for ( int i = 3; i * i <= N; i++ )
{
if ( mark [i] )
{
for ( int j = i * i; j < N; j += 2 * i )
mark [j] = false;
}
}
primeList.clear ();
primeList.push_back (2);
for ( int i = 3; i < N; i += 2 )
{
if ( mark [i] )
primeList.push_back (i);
}
//printf ("%d\n", primeList.size ());
}
int main ()
{
sieve ();
vector <int> twinPrime;
for ( size_t i = 1; i < primeList.size (); i++ )
{
if ( primeList [i] - primeList [i - 1] == 2 )
twinPrime.push_back (primeList [i - 1]);
}
int t;
scanf("%d",&t);
int s;
while ( t-- )
{
scanf("%d",&s);
printf ("%d %d\n", twinPrime [s - 1], twinPrime [s - 1] + 2);
}
return 0;
}
Here is a procedure that could answer your question:
Prime numbers that, when divided by 3, have equal quotients when corrected to decimal 0 (zero) are Twin Primes.
This can be written as
For any pair of prime numbers Px, Py, if [Px/3, 0] = [Py/3, 0] then Px and Py are Prime Twins.
The basis for this is that if prime numbers differ by 2, then dividing the all the prime numbers of interest will yield unique equal quotients when the quotients are corrected to decimal zero. Primes that are not separated by 2 will not have equal quotients when corrected to decimal zero.
For example:
• 11, 13 when divided by 3 will yield unique the unique quotient of 4 when the quotient is corrected to decimal zero.
• 17, 19 when divided by 3 will yield the unique quotient of 6 when the quotient is corrected to decimal zero.
• 29, 31 when divided by 3 will yield the unique quotient of 10 when the quotient is corrected to decimal zero.
Etc.
Below is a simple procedure using Excel to:
• Find prime twins from any list of primes
• Find twin primes in any range of primes
• Find the largest prime twin prime
• Find gaps between twin primes
Import Kutools into Excel
List prime numbers of interest into column 1.
Insert divisor 3 in column 2 - fill down to the level of the largest prime on the list in column 1.
Divide the first row of column 1 by the first row of column 2 and place the quotient in column 3
Fill down column 3 to the level of the largest prime number on the list in column 1.
Correct to zero decimal. Keep the numbers column 3 (quotients) selected.
From “Conditional formatting’ - Select "duplicate values" from the menu
Go to Kutools and select 'to actual' - This will highlight the cells of all the twin pairs scattered in the Quotient column 3.
Select the quotients in column 3
Select 'Sort and Filter' in Excel
Select 'Custom Sort'
Fill in the menu (For values chose the highlighted color in the quotient column) and and click ‘OK”.
The twin primes will be grouped together in the column.
This list can then be used to find the gaps between primes.
To find the largest twin prime use the above procedure with a range of the largest known prime into column 1 (e.g. the highest 10k primes).
If a prime twin is not found in this range, then go to the next lowest range until a twin prime is found. This will be the largest twin prime.
Hope this helps.
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.