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Performance of the KMP string matching algorithm

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• Review of the KMP algorithm
  • The KMP Algorithm:

    int KMP(String T, String P) { f() = KMP_failure_function(P); i = 0; j = 0; n = T.length(); m = P.length(); while ( i < n ) { if ( P[j] == T[i] ) { /* ------------------------ Check if we found P ------------------------ */ if ( j == (m-1) ) { return( i - (m-1) ); // Found P in T ! } /* ------------------------ Not yet, keep going... ------------------------ */ i++; j++; } else { /* --------------------------- Fail to match at P[j] ---------------------------- */ if ( j == 0 ) { /* ------------------------------------------------- First character in P ==> try next character in T ------------------------------------------------- */ i++; } else { /* ----------------------------------------------------- There is a prefix matched ==> go to the max matching suffix of the prefix ----------------------------------------------------- */ j = f(j-1); } } } return(-1); // No match found... }

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• Performance of the KMP string matching algorithm
  • The usage of the variables i and j in the KMP algorithm:

    0123456789012345678901234567801 (ruler) i | v T: xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx P: yyyyyyy ^ | j 0123456 (ruler) i = index of the character in text T being considered j = index of the character in pattern P being considered The KMP algorithm is testing whether: Ti == Pj

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  • Define:   k = i − j

    0123456789012345678901234567801 (ruler) i | v T: xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx P: yyyyyyy ^ ^ | | | j 0123456 (ruler) | k = i - j k = index of the first letter of T in the string matching process

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  • Consider the body of the while loop in the KMP algorithm:

    while ( i < n ) { if ( P[j] == T[i] ) { /* ------------------------ Check if we found P ------------------------ */ if ( j == (m-1) ) { return( i - (m-1) ); // Found P in T ! } /* ------------------------ Not yet, keep going... ------------------------ */ i++; ********** j++; ********** } else { /* --------------------------- Fail to match at P[j] ---------------------------- */ if ( j == 0 ) { /* ------------------------------------------------- First character in P ==> try next character in T ------------------------------------------------- */ i++; ********** } else { /* ----------------------------------------------------- There is a prefix matched ==> go to the max matching suffix of the prefix ----------------------------------------------------- */ j = f(j-1); ********** } } }

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  • In each iteration of the while loop, exactly one of these 3 things will happen:

    i++ and j++             ⇒       i++, k unchanged             i++                            ⇒       i++, k++ j = f(j−1)                  ⇒       i unchanged, k++    (because f(j−1) < j)

    Summary:

    In each iteration of the while loop, at least one of the variables k or i is increased by at least 1

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  • The range of value of the variables k and i = [0..n]

    Behold:

    0123456789012345678901234567801 (ruler) i | v T: xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx P: yyyyyyy ^ ^ | | | j 0123456 (ruler) | k = i - j <------------------------------> range(i) = n characters in T range(k) = n characters in T

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  • Therefore:

    You can increase i ≤ n times          and               You can increase k ≤ n times

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  • Because each time through the while loop, we will eitehr increase i by ≤ 1 or increase k by ≤ 1, the maximum # times that the while loop is executed is:

    # time while loop can be executed ≤ 2 × n

    (Otherwise, we will increase one of the variable i or k by more than n !!!)

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  • Therefore:

    Running time of the KMP algorithm = O(n)

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• WAIT:   what about the failure function computation ???
  • Before the KMP algorithm can be used, we must first compute the failure function f(i)

  • The computation of the failure function f(i) is part of the running time of the KMP algorithm

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• Review of the KMP failure function
  • The KMP failure function computation algorithm:

    int KMP_failure_function(String P) { int i, j, m; int f[] = new int[P.length() + 1]; m = P.length(); f[0] = 0; // See KMP alg. for explanation (click here) i = 1; // Next value of f[] to compute j = 0; /* --------------------------------------- We are at this situation: p1 (i=1) p0 p1 .... pm-1 (j=0) --------------------------------------- */ while ( i < m ) { if ( P[j] == P[i] ) { /* ------------------------ Case 1: ------------------------ */ f[i] = j; // ******* Case 1 /* ------------------------ Next position... ------------------------ */ i++; j++; } else { /* --------------------------- Case 2: does not match ---------------------------- */ if ( j == 0 ) { /* ------------------------------------------------- No match possible ------------------------------------------------- */ f[i] = 0; // ********* i++; } else { /* ----------------------------------------------------- There is a prefix matched ==> go to the max matching suffix of the prefix and CONTINUE the search ----------------------------------------------------- */ j = f(j-1); // Jump and continue search... } } } }

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• Running time of the failure function computation
  • Using the same argument (but for a text string of length m), we conclude that:

    # time while loop can be executed ≤ 2 × m

    (Otherwise, we will increase one of the variable i or k by more than n !!!)

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  • Therefore:

    Running time of the failure function computation = O(m)

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• Total running time of the KMP algorithm
  • Total running time of KMP:

    O(m + n)