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Rolling Hash

Rolling hash is one common trick used to increase efficiency of substring comparisons by compressing (or hashing) a string into a integer. After this step, we can compare two strings directly without comparing each chars. So the efficiency can be increased from O(N) to O(1).

So how to implement the rolling hash?

First we need to choose a base for the expansion and a modulo to mod. The basic formula is (suppose the window is n, and the rolling direction is from left to right),

HashVal = (A1*p^(n-1) + A2*p^(n-2) + ... + An-1*p^1 + An*p^0)%mod

where HashVal is the hash value, Ai is the ith element, p is the base, and mod is the modulo.

To avoid collision as much as we can, p and modulo usually need to be large prime numbers.

One corner case is that the base order in the above formula cannot be reversed. Or to be more clear, if the rolling direction is from left to right in an array, the first element should be in the highest order of the base, or times p^(n-1), and the last element times p^0. Then when rolling to the next element, we just need to do two steps:

1. continue to adding the next element as previously

HashVal = (HashVal * p % mod + An+1 * p^0) % mod

2. remove the first element (since it is out of the window n now)

HashVal = (HashVal - A1 * p^n + mod) % mod

If we use a reversed order of the base (in this case, A1 would time p^0 and An * p^(n-1)), we need to divide p instead of times when rolling to the next element, which will lead errors in the module step. More details can be found in the first example in the Question List below.

Question List


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