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 came across the following question:
Using rand() function, generate a number with expected value k. Options are:
1)
int GetRandom(int k)
{
v=0;
while(rand()<1.0f/(float)k)
v++;
return v;
}
2)
int GetRandom(int k)
{
v=0;
while(rand()<(1-1.0f/(float)k))
v++;
return v;
}
3)
int GetRandom(int k)
{
v=0;
while(rand() > (1-1.0f/(float)(k+1)))
v++;
return v;
}
1) seemed like the correct answer. Examining the outcome for specific values of k seems to indicate this is the not the case. (I set k=3. The frequency distribution of values for 100000 trials can be seen in the image below )
How would one do this ?
The question is somewhat similar to this one.
You want (2). This samples a Geometric Distribution (link) with mean k.
A geometric distribution represents an experiment of this kind:
A certain event happens repeatedly, with an outcome that is either 0 or 1
The outcome of an event is 1 with probability p and 0 with probability 1-p
What is the index of the first event with an outcome of 1?
So if X ~ G(p), where X is a random variable and p is the probability above, then X represents "What is the index of the first event with an outcome of 1?" The expectation is E[X] = 1/p.
Given this information it should now be clear that the following represents a sampling of the random variable X with p = 1/k (and is equivalent to (2)).
int Sample(int k)
{
int v = 1;
while (true)
{
//outcome is true with probability p = 1/k
bool outcome = rand() < 1 / (double)k;
if (outcome)
return v;
else
v++;
}
}
Be aware that looking at the peak (mode) and expectation of the distribution are not the same thing. The peak of the geometric distribution is always going to be at 1!
I'm working on a project for fun and I need an algorithm to do as follows:
Generate a list of numbers of Length n which add up to x
I would settle for list of integers, but ideally, I would like to be left with a set of floating point numbers.
I would be very surprised if this problem wasn't heavily studied, but I'm not sure what to look for.
I've tackled similar problems in the past, but this one is decidedly different in nature. Before I've generated different combinations of a list of numbers that will add up to x. I'm sure that I could simply bruteforce this problem but that hardly seems like the ideal solution.
Anyone have any idea what this may be called, or how to approach it? Thanks all!
Edit: To clarify, I mean that the list should be length N while the numbers themselves can be of any size.
edit2: Sorry for my improper use of 'set', I was using it as a catch all term for a list or an array. I understand that it was causing confusion, my apologies.
This is how to do it in Python
import random
def random_values_with_prescribed_sum(n, total):
x = [random.random() for i in range(n)]
k = total / sum(x)
return [v * k for v in x]
Basically you pick n random numbers, compute their sum and compute a scale factor so that the sum will be what you want it to be.
Note that this approach will not produce "uniform" slices, i.e. the distribution you will get will tend to be more "egalitarian" than it should be if it was picked at random among all distribution with the given sum.
To see the reason you can just picture what the algorithm does in the case of two numbers with a prescribed sum (e.g. 1):
The point P is a generic point obtained by picking two random numbers and it will be uniform inside the square [0,1]x[0,1]. The point Q is the point obtained by scaling P so that the sum is required to be 1. As it's clear from the picture the points close to the center of the have an higher probability; for example the exact center of the squares will be found by projecting any point on the diagonal (0,0)-(1,1), while the point (0, 1) will be found projecting only points from (0,0)-(0,1)... the diagonal length is sqrt(2)=1.4142... while the square side is only 1.0.
Actually, you need to generate a partition of x into n parts. This is usually done the in following way: The partition of x into n non-negative parts can be represented in the following way: reserve n + x free places, put n borders to some arbitrary places, and stones to the rest. The stone groups add up to x, thus the number of possible partitions is the binomial coefficient (n + x \atop n).
So your algorithm could be as follows: choose an arbitrary n-subset of (n + x)-set, it determines uniquely a partition of x into n parts.
In Knuth's TAOCP the chapter 3.4.2 discusses random sampling. See Algortihm S there.
Algorithm S: (choose n arbitrary records from total of N)
t = 0, m = 0;
u = random, uniformly distributed on (0, 1)
if (N - t)*u >= n - m, skip t-th record and increase t by 1; otherwise include t-th record in the sample, increase m and t by 1
if M < n, return to 2, otherwise, algorithm finished
The solution for non-integers is algorithmically trivial: you just select arbitrary n numbers that don't sum up to 0, and norm them by their sum.
If you want to sample uniformly in the region of N-1-dimensional space defined by x1 + x2 + ... + xN = x, then you're looking at a special case of sampling from a Dirichlet distribution. The sampling procedure is a little more involved than generating uniform deviates for the xi. Here's one way to do it, in Python:
xs = [random.gammavariate(1,1) for a in range(N)]
xs = [x*v/sum(xs) for v in xs]
If you don't care too much about the sampling properties of your results, you can just generate uniform deviates and correct their sum afterwards.
Here is a version of the above algorithm in Javascript
function getRandomArbitrary(min, max) {
return Math.random() * (max - min) + min;
};
function getRandomArray(min, max, n) {
var arr = [];
for (var i = 0, l = n; i < l; i++) {
arr.push(getRandomArbitrary(min, max))
};
return arr;
};
function randomValuesPrescribedSum(min, max, n, total) {
var arr = getRandomArray(min, max, n);
var sum = arr.reduce(function(pv, cv) { return pv + cv; }, 0);
var k = total/sum;
var delays = arr.map(function(x) { return k*x; })
return delays;
};
You can call it with
var myarray = randomValuesPrescribedSum(0,1,3,3);
And then check it with
var sum = myarray.reduce(function(pv, cv) { return pv + cv;},0);
This code does a reasonable job. I think it produces a different distribution than 6502's answer, but I am not sure which is better or more natural. Certainly his code is clearer/nicer.
import random
def parts(total_sum, num_parts):
points = [random.random() for i in range(num_parts-1)]
points.append(0)
points.append(1)
points.sort()
ret = []
for i in range(1, len(points)):
ret.append((points[i] - points[i-1]) * total_sum)
return ret
def test(total_sum, num_parts):
ans = parts(total_sum, num_parts)
assert abs(sum(ans) - total_sum) < 1e-7
print ans
test(5.5, 3)
test(10, 1)
test(10, 5)
In python:
a: create a list of (random #'s 0 to 1) times total; append 0 and total to the list
b: sort the list, measure the distance between each element
c: round the list elements
import random
import time
TOTAL = 15
PARTS = 4
PLACES = 3
def random_sum_split(parts, total, places):
a = [0, total] + [random.random()*total for i in range(parts-1)]
a.sort()
b = [(a[i] - a[i-1]) for i in range(1, (parts+1))]
if places == None:
return b
else:
b.pop()
c = [round(x, places) for x in b]
c.append(round(total-sum(c), places))
return c
def tick():
if info.tick == 1:
start = time.time()
alpha = random_sum_split(PARTS, TOTAL, PLACES)
end = time.time()
log('alpha: %s' % alpha)
log('total: %.7f' % sum(alpha))
log('parts: %s' % PARTS)
log('places: %s' % PLACES)
log('elapsed: %.7f' % (end-start))
yields:
[2014-06-13 01:00:00] alpha: [0.154, 3.617, 6.075, 5.154]
[2014-06-13 01:00:00] total: 15.0000000
[2014-06-13 01:00:00] parts: 4
[2014-06-13 01:00:00] places: 3
[2014-06-13 01:00:00] elapsed: 0.0005839
to the best of my knowledge this distribution is uniform
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.
Selecting without any weights (equal probabilities) is beautifully described here.
I was wondering if there is a way to convert this approach to a weighted one.
I am also interested in other approaches as well.
Update: Sampling without replacement
If the sampling is with replacement, you can use this algorithm (implemented here in Python):
import random
items = [(10, "low"),
(100, "mid"),
(890, "large")]
def weighted_sample(items, n):
total = float(sum(w for w, v in items))
i = 0
w, v = items[0]
while n:
x = total * (1 - random.random() ** (1.0 / n))
total -= x
while x > w:
x -= w
i += 1
w, v = items[i]
w -= x
yield v
n -= 1
This is O(n + m) where m is the number of items.
Why does this work? It is based on the following algorithm:
def n_random_numbers_decreasing(v, n):
"""Like reversed(sorted(v * random() for i in range(n))),
but faster because we avoid sorting."""
while n:
v *= random.random() ** (1.0 / n)
yield v
n -= 1
The function weighted_sample is just this algorithm fused with a walk of the items list to pick out the items selected by those random numbers.
This in turn works because the probability that n random numbers 0..v will all happen to be less than z is P = (z/v)n. Solve for z, and you get z = vP1/n. Substituting a random number for P picks the largest number with the correct distribution; and we can just repeat the process to select all the other numbers.
If the sampling is without replacement, you can put all the items into a binary heap, where each node caches the total of the weights of all items in that subheap. Building the heap is O(m). Selecting a random item from the heap, respecting the weights, is O(log m). Removing that item and updating the cached totals is also O(log m). So you can pick n items in O(m + n log m) time.
(Note: "weight" here means that every time an element is selected, the remaining possibilities are chosen with probability proportional to their weights. It does not mean that elements appear in the output with a likelihood proportional to their weights.)
Here's an implementation of that, plentifully commented:
import random
class Node:
# Each node in the heap has a weight, value, and total weight.
# The total weight, self.tw, is self.w plus the weight of any children.
__slots__ = ['w', 'v', 'tw']
def __init__(self, w, v, tw):
self.w, self.v, self.tw = w, v, tw
def rws_heap(items):
# h is the heap. It's like a binary tree that lives in an array.
# It has a Node for each pair in `items`. h[1] is the root. Each
# other Node h[i] has a parent at h[i>>1]. Each node has up to 2
# children, h[i<<1] and h[(i<<1)+1]. To get this nice simple
# arithmetic, we have to leave h[0] vacant.
h = [None] # leave h[0] vacant
for w, v in items:
h.append(Node(w, v, w))
for i in range(len(h) - 1, 1, -1): # total up the tws
h[i>>1].tw += h[i].tw # add h[i]'s total to its parent
return h
def rws_heap_pop(h):
gas = h[1].tw * random.random() # start with a random amount of gas
i = 1 # start driving at the root
while gas >= h[i].w: # while we have enough gas to get past node i:
gas -= h[i].w # drive past node i
i <<= 1 # move to first child
if gas >= h[i].tw: # if we have enough gas:
gas -= h[i].tw # drive past first child and descendants
i += 1 # move to second child
w = h[i].w # out of gas! h[i] is the selected node.
v = h[i].v
h[i].w = 0 # make sure this node isn't chosen again
while i: # fix up total weights
h[i].tw -= w
i >>= 1
return v
def random_weighted_sample_no_replacement(items, n):
heap = rws_heap(items) # just make a heap...
for i in range(n):
yield rws_heap_pop(heap) # and pop n items off it.
If the sampling is with replacement, use the roulette-wheel selection technique (often used in genetic algorithms):
sort the weights
compute the cumulative weights
pick a random number in [0,1]*totalWeight
find the interval in which this number falls into
select the elements with the corresponding interval
repeat k times
If the sampling is without replacement, you can adapt the above technique by removing the selected element from the list after each iteration, then re-normalizing the weights so that their sum is 1 (valid probability distribution function)
I know this is a very old question, but I think there's a neat trick to do this in O(n) time if you apply a little math!
The exponential distribution has two very useful properties.
Given n samples from different exponential distributions with different rate parameters, the probability that a given sample is the minimum is equal to its rate parameter divided by the sum of all rate parameters.
It is "memoryless". So if you already know the minimum, then the probability that any of the remaining elements is the 2nd-to-min is the same as the probability that if the true min were removed (and never generated), that element would have been the new min. This seems obvious, but I think because of some conditional probability issues, it might not be true of other distributions.
Using fact 1, we know that choosing a single element can be done by generating these exponential distribution samples with rate parameter equal to the weight, and then choosing the one with minimum value.
Using fact 2, we know that we don't have to re-generate the exponential samples. Instead, just generate one for each element, and take the k elements with lowest samples.
Finding the lowest k can be done in O(n). Use the Quickselect algorithm to find the k-th element, then simply take another pass through all elements and output all lower than the k-th.
A useful note: if you don't have immediate access to a library to generate exponential distribution samples, it can be easily done by: -ln(rand())/weight
I've done this in Ruby
https://github.com/fl00r/pickup
require 'pickup'
pond = {
"selmon" => 1,
"carp" => 4,
"crucian" => 3,
"herring" => 6,
"sturgeon" => 8,
"gudgeon" => 10,
"minnow" => 20
}
pickup = Pickup.new(pond, uniq: true)
pickup.pick(3)
#=> [ "gudgeon", "herring", "minnow" ]
pickup.pick
#=> "herring"
pickup.pick
#=> "gudgeon"
pickup.pick
#=> "sturgeon"
If you want to generate large arrays of random integers with replacement, you can use piecewise linear interpolation. For example, using NumPy/SciPy:
import numpy
import scipy.interpolate
def weighted_randint(weights, size=None):
"""Given an n-element vector of weights, randomly sample
integers up to n with probabilities proportional to weights"""
n = weights.size
# normalize so that the weights sum to unity
weights = weights / numpy.linalg.norm(weights, 1)
# cumulative sum of weights
cumulative_weights = weights.cumsum()
# piecewise-linear interpolating function whose domain is
# the unit interval and whose range is the integers up to n
f = scipy.interpolate.interp1d(
numpy.hstack((0.0, weights)),
numpy.arange(n + 1), kind='linear')
return f(numpy.random.random(size=size)).astype(int)
This is not effective if you want to sample without replacement.
Here's a Go implementation from geodns:
package foo
import (
"log"
"math/rand"
)
type server struct {
Weight int
data interface{}
}
func foo(servers []server) {
// servers list is already sorted by the Weight attribute
// number of items to pick
max := 4
result := make([]server, max)
sum := 0
for _, r := range servers {
sum += r.Weight
}
for si := 0; si < max; si++ {
n := rand.Intn(sum + 1)
s := 0
for i := range servers {
s += int(servers[i].Weight)
if s >= n {
log.Println("Picked record", i, servers[i])
sum -= servers[i].Weight
result[si] = servers[i]
// remove the server from the list
servers = append(servers[:i], servers[i+1:]...)
break
}
}
}
return result
}
If you want to pick x elements from a weighted set without replacement such that elements are chosen with a probability proportional to their weights:
import random
def weighted_choose_subset(weighted_set, count):
"""Return a random sample of count elements from a weighted set.
weighted_set should be a sequence of tuples of the form
(item, weight), for example: [('a', 1), ('b', 2), ('c', 3)]
Each element from weighted_set shows up at most once in the
result, and the relative likelihood of two particular elements
showing up is equal to the ratio of their weights.
This works as follows:
1.) Line up the items along the number line from [0, the sum
of all weights) such that each item occupies a segment of
length equal to its weight.
2.) Randomly pick a number "start" in the range [0, total
weight / count).
3.) Find all the points "start + n/count" (for all integers n
such that the point is within our segments) and yield the set
containing the items marked by those points.
Note that this implementation may not return each possible
subset. For example, with the input ([('a': 1), ('b': 1),
('c': 1), ('d': 1)], 2), it may only produce the sets ['a',
'c'] and ['b', 'd'], but it will do so such that the weights
are respected.
This implementation only works for nonnegative integral
weights. The highest weight in the input set must be less
than the total weight divided by the count; otherwise it would
be impossible to respect the weights while never returning
that element more than once per invocation.
"""
if count == 0:
return []
total_weight = 0
max_weight = 0
borders = []
for item, weight in weighted_set:
if weight < 0:
raise RuntimeError("All weights must be positive integers")
# Scale up weights so dividing total_weight / count doesn't truncate:
weight *= count
total_weight += weight
borders.append(total_weight)
max_weight = max(max_weight, weight)
step = int(total_weight / count)
if max_weight > step:
raise RuntimeError(
"Each weight must be less than total weight / count")
next_stop = random.randint(0, step - 1)
results = []
current = 0
for i in range(count):
while borders[current] <= next_stop:
current += 1
results.append(weighted_set[current][0])
next_stop += step
return results
In the question you linked to, Kyle's solution would work with a trivial generalization.
Scan the list and sum the total weights. Then the probability to choose an element should be:
1 - (1 - (#needed/(weight left)))/(weight at n). After visiting a node, subtract it's weight from the total. Also, if you need n and have n left, you have to stop explicitly.
You can check that with everything having weight 1, this simplifies to kyle's solution.
Edited: (had to rethink what twice as likely meant)
This one does exactly that with O(n) and no excess memory usage. I believe this is a clever and efficient solution easy to port to any language. The first two lines are just to populate sample data in Drupal.
function getNrandomGuysWithWeight($numitems){
$q = db_query('SELECT id, weight FROM theTableWithTheData');
$q = $q->fetchAll();
$accum = 0;
foreach($q as $r){
$accum += $r->weight;
$r->weight = $accum;
}
$out = array();
while(count($out) < $numitems && count($q)){
$n = rand(0,$accum);
$lessaccum = NULL;
$prevaccum = 0;
$idxrm = 0;
foreach($q as $i=>$r){
if(($lessaccum == NULL) && ($n <= $r->weight)){
$out[] = $r->id;
$lessaccum = $r->weight- $prevaccum;
$accum -= $lessaccum;
$idxrm = $i;
}else if($lessaccum){
$r->weight -= $lessaccum;
}
$prevaccum = $r->weight;
}
unset($q[$idxrm]);
}
return $out;
}
I putting here a simple solution for picking 1 item, you can easily expand it for k items (Java style):
double random = Math.random();
double sum = 0;
for (int i = 0; i < items.length; i++) {
val = items[i];
sum += val.getValue();
if (sum > random) {
selected = val;
break;
}
}
I have implemented an algorithm similar to Jason Orendorff's idea in Rust here. My version additionally supports bulk operations: insert and remove (when you want to remove a bunch of items given by their ids, not through the weighted selection path) from the data structure in O(m + log n) time where m is the number of items to remove and n the number of items in stored.
Sampling wihout replacement with recursion - elegant and very short solution in c#
//how many ways we can choose 4 out of 60 students, so that every time we choose different 4
class Program
{
static void Main(string[] args)
{
int group = 60;
int studentsToChoose = 4;
Console.WriteLine(FindNumberOfStudents(studentsToChoose, group));
}
private static int FindNumberOfStudents(int studentsToChoose, int group)
{
if (studentsToChoose == group || studentsToChoose == 0)
return 1;
return FindNumberOfStudents(studentsToChoose, group - 1) + FindNumberOfStudents(studentsToChoose - 1, group - 1);
}
}
I just spent a few hours trying to get behind the algorithms underlying sampling without replacement out there and this topic is more complex than I initially thought. That's exciting! For the benefit of a future readers (have a good day!) I document my insights here including a ready to use function which respects the given inclusion probabilities further below. A nice and quick mathematical overview of the various methods can be found here: Tillé: Algorithms of sampling with equal or unequal probabilities. For example Jason's method can be found on page 46. The caveat with his method is that the weights are not proportional to the inclusion probabilities as also noted in the document. Actually, the i-th inclusion probabilities can be recursively computed as follows:
def inclusion_probability(i, weights, k):
"""
Computes the inclusion probability of the i-th element
in a randomly sampled k-tuple using Jason's algorithm
(see https://stackoverflow.com/a/2149533/7729124)
"""
if k <= 0: return 0
cum_p = 0
for j, weight in enumerate(weights):
# compute the probability of j being selected considering the weights
p = weight / sum(weights)
if i == j:
# if this is the target element, we don't have to go deeper,
# since we know that i is included
cum_p += p
else:
# if this is not the target element, than we compute the conditional
# inclusion probability of i under the constraint that j is included
cond_i = i if i < j else i-1
cond_weights = weights[:j] + weights[j+1:]
cond_p = inclusion_probability(cond_i, cond_weights, k-1)
cum_p += p * cond_p
return cum_p
And we can check the validity of the function above by comparing
In : for i in range(3): print(i, inclusion_probability(i, [1,2,3], 2))
0 0.41666666666666663
1 0.7333333333333333
2 0.85
to
In : import collections, itertools
In : sample_tester = lambda f: collections.Counter(itertools.chain(*(f() for _ in range(10000))))
In : sample_tester(lambda: random_weighted_sample_no_replacement([(1,'a'),(2,'b'),(3,'c')],2))
Out: Counter({'a': 4198, 'b': 7268, 'c': 8534})
One way - also suggested in the document above - to specify the inclusion probabilities is to compute the weights from them. The whole complexity of the question at hand stems from the fact that one cannot do that directly since one basically has to invert the recursion formula, symbolically I claim this is impossible. Numerically it can be done using all kind of methods, e.g. Newton's method. However the complexity of inverting the Jacobian using plain Python becomes unbearable quickly, I really recommend looking into numpy.random.choice in this case.
Luckily there is method using plain Python which might or might not be sufficiently performant for your purposes, it works great if there aren't that many different weights. You can find the algorithm on page 75&76. It works by splitting up the sampling process into parts with the same inclusion probabilities, i.e. we can use random.sample again! I am not going to explain the principle here since the basics are nicely presented on page 69. Here is the code with hopefully a sufficient amount of comments:
def sample_no_replacement_exact(items, k, best_effort=False, random_=None, ε=1e-9):
"""
Returns a random sample of k elements from items, where items is a list of
tuples (weight, element). The inclusion probability of an element in the
final sample is given by
k * weight / sum(weights).
Note that the function raises if a inclusion probability cannot be
satisfied, e.g the following call is obviously illegal:
sample_no_replacement_exact([(1,'a'),(2,'b')],2)
Since selecting two elements means selecting both all the time,
'b' cannot be selected twice as often as 'a'. In general it can be hard to
spot if the weights are illegal and the function does *not* always raise
an exception in that case. To remedy the situation you can pass
best_effort=True which redistributes the inclusion probability mass
if necessary. Note that the inclusion probabilities will change
if deemed necessary.
The algorithm is based on the splitting procedure on page 75/76 in:
http://www.eustat.eus/productosServicios/52.1_Unequal_prob_sampling.pdf
Additional information can be found here:
https://stackoverflow.com/questions/2140787/
:param items: list of tuples of type weight,element
:param k: length of resulting sample
:param best_effort: fix inclusion probabilities if necessary,
(optional, defaults to False)
:param random_: random module to use (optional, defaults to the
standard random module)
:param ε: fuzziness parameter when testing for zero in the context
of floating point arithmetic (optional, defaults to 1e-9)
:return: random sample set of size k
:exception: throws ValueError in case of bad parameters,
throws AssertionError in case of algorithmic impossibilities
"""
# random_ defaults to the random submodule
if not random_:
random_ = random
# special case empty return set
if k <= 0:
return set()
if k > len(items):
raise ValueError("resulting tuple length exceeds number of elements (k > n)")
# sort items by weight
items = sorted(items, key=lambda item: item[0])
# extract the weights and elements
weights, elements = list(zip(*items))
# compute the inclusion probabilities (short: π) of the elements
scaling_factor = k / sum(weights)
π = [scaling_factor * weight for weight in weights]
# in case of best_effort: if a inclusion probability exceeds 1,
# try to rebalance the probabilities such that:
# a) no probability exceeds 1,
# b) the probabilities still sum to k, and
# c) the probability masses flow from top to bottom:
# [0.2, 0.3, 1.5] -> [0.2, 0.8, 1]
# (remember that π is sorted)
if best_effort and π[-1] > 1 + ε:
# probability mass we still we have to distribute
debt = 0.
for i in reversed(range(len(π))):
if π[i] > 1.:
# an 'offender', take away excess
debt += π[i] - 1.
π[i] = 1.
else:
# case π[i] < 1, i.e. 'save' element
# maximum we can transfer from debt to π[i] and still not
# exceed 1 is computed by the minimum of:
# a) 1 - π[i], and
# b) debt
max_transfer = min(debt, 1. - π[i])
debt -= max_transfer
π[i] += max_transfer
assert debt < ε, "best effort rebalancing failed (impossible)"
# make sure we are talking about probabilities
if any(not (0 - ε <= π_i <= 1 + ε) for π_i in π):
raise ValueError("inclusion probabilities not satisfiable: {}" \
.format(list(zip(π, elements))))
# special case equal probabilities
# (up to fuzziness parameter, remember that π is sorted)
if π[-1] < π[0] + ε:
return set(random_.sample(elements, k))
# compute the two possible lambda values, see formula 7 on page 75
# (remember that π is sorted)
λ1 = π[0] * len(π) / k
λ2 = (1 - π[-1]) * len(π) / (len(π) - k)
λ = min(λ1, λ2)
# there are two cases now, see also page 69
# CASE 1
# with probability λ we are in the equal probability case
# where all elements have the same inclusion probability
if random_.random() < λ:
return set(random_.sample(elements, k))
# CASE 2:
# with probability 1-λ we are in the case of a new sample without
# replacement problem which is strictly simpler,
# it has the following new probabilities (see page 75, π^{(2)}):
new_π = [
(π_i - λ * k / len(π))
/
(1 - λ)
for π_i in π
]
new_items = list(zip(new_π, elements))
# the first few probabilities might be 0, remove them
# NOTE: we make sure that floating point issues do not arise
# by using the fuzziness parameter
while new_items and new_items[0][0] < ε:
new_items = new_items[1:]
# the last few probabilities might be 1, remove them and mark them as selected
# NOTE: we make sure that floating point issues do not arise
# by using the fuzziness parameter
selected_elements = set()
while new_items and new_items[-1][0] > 1 - ε:
selected_elements.add(new_items[-1][1])
new_items = new_items[:-1]
# the algorithm reduces the length of the sample problem,
# it is guaranteed that:
# if λ = λ1: the first item has probability 0
# if λ = λ2: the last item has probability 1
assert len(new_items) < len(items), "problem was not simplified (impossible)"
# recursive call with the simpler sample problem
# NOTE: we have to make sure that the selected elements are included
return sample_no_replacement_exact(
new_items,
k - len(selected_elements),
best_effort=best_effort,
random_=random_,
ε=ε
) | selected_elements
Example:
In : sample_no_replacement_exact([(1,'a'),(2,'b'),(3,'c')],2)
Out: {'b', 'c'}
In : import collections, itertools
In : sample_tester = lambda f: collections.Counter(itertools.chain(*(f() for _ in range(10000))))
In : sample_tester(lambda: sample_no_replacement_exact([(1,'a'),(2,'b'),(3,'c'),(4,'d')],2))
Out: Counter({'a': 2048, 'b': 4051, 'c': 5979, 'd': 7922})
The weights sum up to 10, hence the inclusion probabilities compute to: a → 20%, b → 40%, c → 60%, d → 80%. (Sum: 200% = k.) It works!
Just one word of caution for the productive use of this function, it can be very hard to spot illegal inputs for the weights. An obvious illegal example is
In: sample_no_replacement_exact([(1,'a'),(2,'b')],2)
ValueError: inclusion probabilities not satisfiable: [(0.6666666666666666, 'a'), (1.3333333333333333, 'b')]
b cannot appear twice as often as a since both have to be always be selected. There are more subtle examples. To avoid an exception in production just use best_effort=True, which rebalances the inclusion probability mass such that we have always a valid distribution. Obviously this might change the inclusion probabilities.
I used a associative map (weight,object). for example:
{
(10,"low"),
(100,"mid"),
(10000,"large")
}
total=10110
peek a random number between 0 and 'total' and iterate over the keys until this number fits in a given range.