The Z Algorithm: Step-by-Step

Here’s a detailed breakdown of how the Z algorithm works:

• Initialization:
  • Start with the entire string S, and initialize the Z-array Z with zeroes.
  • Set the variables L and R to 0. These variables will define a window in S where S[L:R+1] matches the prefix of S.
• Iterate through the string: For each position i in the string S:
  • Case 1: If i > R, then there is no Z-box (a substring matching the prefix of S that starts before i and ends after i).
    • Set L = R = i and extend the window R to the right as long as S[R] == S[R-L].
    • Set Z[i] = R – L and decrement R.
  • Case 2: If i ≤ R, then i falls within a Z-box. Use the previously computed Z-values to determine the value of Z[i]:
    • Sub-case 2a: If Z[i-L] < R – i + 1, then Z[i] = Z[i-L].
    • Sub-case 2b: If Z[i-L] ≥ R – i + 1, then set L = i and extend the window R as long as S[R] == S[R-L]. Set Z[i] = R – L and decrement R.
• Output the Z-array: After processing all positions in the string, the Z-array contains the lengths of the longest substrings starting from each position that match the prefix of S.

The Z algorithm is a powerful string-matching algorithm used to find all occurrences of a pattern within a text. It operates efficiently, with a linear time complexity of O(n+m), where n is the length of the text and m is the length of the pattern. This makes it particularly useful for problems involving large texts. In this article, we’ll explore the Z algorithm, understand its underlying concepts, and learn how to implement it in Python.