What is the Knuth-Morris-Pratt algorithm?

The Knuth-Morris-Pratt (KMP) algorithm is an algorithm that is used to search for a substring (W), in a given string (S), in $O(m+n)$ time (where $m$ and $n$ are the lengths of W and S).

Key idea

The key idea used to achieve this time complexity is to minimize the amount of backtracking when a character of W does not match with that of S. This can only be done if we know two things:

1. Whether or not a proper prefix of W occurs more than once in S ​after at least one character has been correctly found; if it does, it can be skipped when resuming the process of matching after a mismatch.
2. Length of the proper prefix.

Using an array

Before starting the actual algorithm, a one-dimensional array is initialized with the number of characters that can be skipped after a mismatch. arr[i] represents the number of characters that can be skipped when W[i+1] does not match with a character in S.

In the example above, W[4] (i.e., a) did not match with the character c in S (slide 5), so, the matching was resumed from W[1] instead of W[0] (slide 6). This means that character 1 was skipped; therefore, arr[4-1] = 1.

Failure function

arr is initialized using the failure function $f(i)$. This function is based on the fact that:

When a mismatch occurs, all the previous characters match correctly; ​this implies that if a prefix of W occurred in this set of matching characters, then that prefix is also a suffix of W.

In other words, arr[i] will represent the length of the longest proper prefix of W, which is also a proper suffix of W (considering W till the ith index only).

The following is the code used to initialize arr:

m = len(W)
i = 0
j = 1

# No proper prefix for string of length 1:
arr[0] = 0

while j < m:
    if W[i] == W[j]:
        i++
        arr[j] = i
        j++
    # The first character didn't match:
    else if i == 0:
        arr[j] = 0
        j++
    # Mismatch after at least one matching character:
    else:
        i = arr[i - 1]

Algorithm

The following is the KMP algorithm pseudocode used for searching substring W, in string S, in $O(m+n)$:

# Initialising variables:
i = 0
j = 0
m = len(W)
n = len(S)

# Start search:
while (i < m && j < n){
    # Last character matches -> Substring found:
    if (W[i] == S[j] && i == m - 1):
        return true

    # Character matches:
    else if (W[i] == S[j]):
        i++
        j++
    
    # Character didn't match -> Backtrack:
    else:
        i = arr[i - 1]
        if i == 0:
            j++
}

# Substring not found:
return false

Implementation

The algorithm above (given in the form of a pseudocode) along with the failure function, has been implemented in Python: