Confusion Regarding Rolling Hash in Rabin-Karp Algorithm Java

I was trying to understand the Rabin-Karp algorithm here: http://algs4.cs.princeton.edu/53substring/RabinKarp.java.html.

I have looked through various articles and I now know that the general form of a polynomial hash is C1*A^k-1+C2*A^k-2+C3*A^k-3. Looking at the code, I understand how they add and subtract the digits in the string.

txtHash = (txtHash + Q - RM*txt.charAt(i-M) % Q) % Q; txtHash = (txtHash*R + txt.charAt(i)) % Q;

Here the program is subtracting the leading digit, multiplying the entire hash and then adding the new digit. However, when I was looking through the function that calculates the hash, it didn’t follow the general form of a polynomial hash. It looked like this:

 private long hash(String key, int M) { 
    long h = 0; 
    for (int j = 0; j < M; j++) 
        h = (R * h + key.charAt(j)) % Q; 
    return h; 
} 

In this function they are multiplying the hash and the radix and then adding the key.charAt(). I would figure the function would be multiplying the key.charAt() with a base that starts out at R^k-1. Then as the for loop continues, the base would divided by R to provide for the decreasing power in the polynomial. Can someone please explain how this function works and how does it generate a hash in the form that I mentioned above? Thanks!

Answer

Suppose hash function need to transfer 3 digits. It would look like :

{digits[0]*R^2+digits[1]*R^1+digits[2]}%Q  
= {(digit[0]*R^1+digits[1])*R+digits[2]}%Q  

This would make hash function mush more easier to calculate.

Then apply to Rabin-Karp Algorithm,
you can see

RM = R^2 %Q;(M=2) 

when you want to move next digit to validate,
you need to delete left most digit and add next digit.

txtHash = {[txtHash - R^2*most_left_digit(equal charAt(i-M))]*R+next_digit(equal charAt(i))}%Q  

It’s the same as

txtHash = (txtHash + Q - RM*txt.charAt(i-M) % Q) % Q; 
txtHash = (txtHash*R + txt.charAt(i)) % Q;

Mod Q every steps prevent overflow.

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