Discuss the concept of rolling hash and how it can be used to optimize dynamic programming solutions.?

Rolling hash is a technique that is frequently used in dynamic programming to optimize the time complexities of algorithms that require string or array manipulation. It can be defined as a hash function that allows for the incremental calculation of a hash value for a string as more characters are added or removed from it.

In a typical dynamic programming solution, we often use a table to store the intermediate results of subproblems. However, precomputing hash values for these subproblems at the beginning of the algorithm can be computationally expensive. Rolling hash offers an alternative to calculating hash values for each subproblem by manipulating previously calculated hash values.

The basic idea behind a rolling hash is to treat a string or an array as a sequence of integers and use a formula to compute the hash value of each contiguous subarray. A common formula for implementing rolling hash is the polynomial hash function.

For example, for a string S of length n, with characters S[0], S[1], , S[n-1], a polynomial hash function can be defined as follows:

hash(S) = (S[0]*p^(n-1) + S[1]*p^(n-2) +  + S[n-1]*p^0) mod m

where p is a prime number and m is a large constant.

The key feature of the polynomial hash function is that it allows for the efficient computation of hash values for a subarray of S[i..j] based on the hash value of the prefix S[0..i-1]. This can be achieved using the following formula:

hash(S[i..j]) = (hash(S[0..j]) - hash(S[0..i-1]) * p^(j-i+1)) mod m

By using this formula, we can compute the hash value for each subarray of S in O(n) time, which is much more efficient than computing the hash value for each subarray from scratch.

Rolling hash can be especially useful in dynamic programming problems that involve comparing substrings or subarrays. For example, a common problem in bioinformatics is the longest common substring problem. Given two strings s and t, the task is to find the longest common substring of s and t.

A brute force solution to this problem would require O(n³) time, where n is the length of the longer string. However, the use of a rolling hash can reduce the time complexity to O(n²logn), making it much more efficient.

Other examples of dynamic programming problems that can be optimized with the use of a rolling hash include the longest palindromic substring problem and the string edit distance problem.

In Java, a simple implementation of rolling hash for strings may look like this:

public long rollingHash(String s) {
    int p = 31;
    int m = 1_000_000_007;
    long hash = 0;
    long powP = 1;
    for (char c : s.toCharArray()) {
        hash = (hash + (c - 'a' + 1) * powP) % m;
        powP = (powP * p) % m;
    }
    return hash;
}

This implementation returns the hash value of a string s using the polynomial hash function with p = 31 and m = 1,000,000,007. The hash value is calculated incrementally by adding the contribution of each character to the hash value using the formula ‘hash = (hash + (c - ’a’ + 1) * powP) % m‘, where c is the character, ’a’ is subtracted to convert the character to a 1-based indexing, and powP is a precomputed value of pⁱ.