Z Algorithm

The Z-algorithm finds occurrences of a "word" W within a main "text string" T in linear time O(|W| + |T|).

Given a string S of length n, the algorithm produces an array, Z where Z[i] represents the longest substring starting from S[i] which is also a prefix of S. Finding Z for the string obtained by concatenating the word, W with a nonce character, say $ followed by the text, T, helps with pattern matching, for if there is some index i such that Z[i] equals the pattern length, then the pattern must be present at that point.

While the Z array can be computed with two nested loops in O(|W| * |T|) time, the following strategy shows how to obtain it in linear time, based on the idea that as we iterate over the letters in the string (index i from 1 to n - 1), we maintain an interval [L, R] which is the interval with maximum R such that 1 ≤ L ≤ i ≤ R and S[L...R] is a prefix that is also a substring (if no such interval exists, just let L = R =  - 1). For i = 1, we can simply compute L and R by comparing S[0...] to S[1...].

Example of Z array

Index            0   1   2   3   4   5   6   7   8   9  10  11 
Text             a   a   b   c   a   a   b   x   a   a   a   z
Z values         X   1   0   0   3   1   0   0   2   2   1   0 

Other examples

str =  a a a a a a
Z[] =  x 5 4 3 2 1

str =  a a b a a c d
Z[] =  x 1 0 2 1 0 0

str =  a b a b a b a b
Z[] =  x 0 6 0 4 0 2 0

Example of Z box

Complexity

• Time: O(|W| + |T|)
• Space: O(|W|)

References

• GeeksForGeeks
• YouTube
• Z Algorithm by Ivan Yurchenko