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Rolling Hash

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Rolling Hash

A rolling hash (also known as recursive hashing) is a hash function where the input is hashed in a window that moves through the input.

Intuition​

When sliding a window of length M across a text of length N, recomputing the hash of the window from scratch would take O(M) time for each position. A rolling hash allows us to compute the hash of the new window from the hash of the previous window in O(1) time by dropping the contribution of the character leaving the window and adding the contribution of the new character entering.

Rabin-Karp Algorithm​

The Rabin-Karp string search algorithm is the most famous application of the rolling hash. It uses a polynomial rolling hash function.

Mathematical Formulation​

If S is a string of length M, the polynomial hash is: H(S) = (S[0] \cdot B^{M-1} + S[1] \cdot B^{M-2} + ... + S[M-1] \cdot B^0) \mod P Where B is the base (typically prime) and P is a large prime modulus.

Complexity Analysis​

  • Time Complexity: O(N + M) on average.
  • Space Complexity: O(1).
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