We talked about pattern search using hashing strategy in class. Suppose we want

Question

We talked about pattern search using hashing strategy in class. Suppose we want to find the first occurrence of a string P1P2...Pk in a long input string A1A2...AN. We can solve this problem by hashing the pattern string, obtaining a hash value Hp, and comparing this value with the hash value formed from A1A2...Ak, A2A3...Ak+1, A3A4...Ak+2, and so on until AN-k+1AN-k+2...AN. If we have a match of hash values, we compare the strings character by character to verify the match. We return the position (in A) if the strings actually do match, and we continue in the unlikely event that the match is false. Design a hash function so that the following property is satisfied: if the hash value of AiAi+1...Ai+k-1 is known, then the hash value of Ai+1Ai+2...Ai+k can be computed in constant time. Write an algorithm which uses your designed hash function in part a) and has the running time 0(k+N) plus the time spent refuting false matches.

Explanation / Answer

We will take the ascii values of letter in computation of hash value of a string and we choose a prime number .

our hase function will be like

          h = ((Ascii(A[i..i+k-1])) mod p where p is a prime number

so since summation function requires k addition so computation of h will take O(k) time.

Note . A[i] denote the ith character and A[i..i+k-1] denotes the character sequence from A[i] to A[i+k-1] of length k.

A).

Suppose we are given the hash value h of A[i]..A[i+k-1] .

then we can compute the hash h' of A[i+1] .. A[i+k]

h' = h - (Ascii(A[i]) mod p ) + (Ascii(A[i+k]) mod p)  

which is similar to

h' = ((Ascii(A[i..i+k-1] mod p )) - (Ascii(A[i]) mod p ) + (Ascii(A[i+k]) mod p)

which is similar to

h' = ((Ascii(A[i+1..i+k] mod p ))

Since we are performing one subtraction and one addition ,this takes constant time.

B)

create an array H of length N-k+1 to store hash value and k is given

1 ) for i = 1 to N-k+1

2 )           compute hash value of A[i..i+k-1] and store in H[i].

Now given S and its hash value h we have to find the index where it matched and refute if match is false.

Match(h,S,A,k)

1 ) found = false;

2 ) for i = 1 to N-k+1 ; do

           if ( h == H[i] )   ; then                                   // if hash value is equal

                   found = stringcmp(S,A[i..i+k-1]);            
                   /*compare two string the string S whose h is given and and A[i..i+k-1]. if the strings are
                       equal the then return index return true otherwise return false. */

                   if ( found == true )

                        return i;

    done

3 ) /* if we reach here the we haven;t found the true occurence of string */

        return -1

bool stringcmp ( S,T ) {

    for i = 1 to k      {

        if ( S[i] != T[i] ) {

           return false;

        }

    }

    return true;                      // if we reach here we havn't found a mismatch.

}        

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