Z algorithm

2016-02-18 2024-05-07 #algorithms #post

String matching is a very common problem, given a text T and a pattern P find the occurrences of P in T. This problem doesn’t need too much introduction.

The most easy (and naive) algorithm to solve this problem is to slide off P through T and see if there is a match. Something like this:

for (int i = 0; i + P.length <= T.length; i++) {
    int j = 0;
    while (P[j] == T[i+j])
    if (j == P.length)
        // We found a match!

However, much more efficient solutions exists to this problem, the Z algorithm is one of them.

The Zi(T) function

The Zi(T) is equal to the length of the longest substring starting at position i > 0 that matches a prefix of T.

For example, let be T = “cabacadcab”, Z3 = 0, Z4 = 2 and Z7 = 3. Obviously Z0 is always equal to the length of the string. See figure .

Z Algorithm example

Z Boxes

If $Z_{i} > 0$ a Z-box is a substring that starts at position $i$ and ends at position $i + Z_{i} - 1$, i.e. the substring that matches the prefix of $T$.

Assume that we have already computed the values of Z upto some k − 1 and now we need to compute Zk. There are four possible cases. In the following pictures l and r denote de start and the end of the last Z-box.

Case 1: $ k $ is out of the last Z-box

Position k is not contained in the last Z-box. We need to compare character by character to find Zk.

Case 1

Case 2: $ k $ is within the last Z-box

We’ll denote this last Z-box as α and as β the box starting at position k and ending at r. Since every Z-box matches a prefix, the following figure depicts our situation.

Case 2

As you can see, k corresponds to position k in the prefix and we already computed Zk and so we can leverage this fact. There are three more cases.

Case 2a: $ Z_{k’} < |\beta| $

In this case Zk = Zk.

Case 2a

Case 2b: $ Z_{k’} > |\beta| $

Let be x the first character that is not contained in the last Z-box and y the first character that is not contained in the prefix, we know one thing, x ≠ y and therefore Zk cannot be greater than Zk. Result Zk = |β|.

Case 2b

Case 2c: $ Z_{k’} = || $

Case 2c

Here we know two things, x ≠ y and y ≠ w. How about x and w? We don’t know, they may be equal or not. In this case it’s necessary to verify.

Okay, here is an implementation to complement the explanation:

Z[0] = n;
int l = 0, r = 0;
for (int k = 1; k < n; k++) {
    if (r < k) {
        l = r = k;
        while (S[r] == S[r-l])
        Z[k] = r - l;
    } else {
        int b = r - k;
        int j = k - l; // j is k'
        if (Z[j] < b) {
            Z[k] = Z[j];
        } else if (Z[j] > b) {
            Z[k] = b;
        } else {
            l = k;
            r = k + b;
            while (S[r] == S[r-l])
            Z[k] = r - l;

Since l and r never decrease the complexity of this algorithm is O(n).

String matching using the Z function

Now that we know how to compute the Z function let’s see how to use it to find the occurrences of a pattern P in a text T. The idea is easy:

  1. Concatenate P with T and compute the Z function in the resulting string (S = P + T )
  2. There is an occurrence of P that start at position i >  = |P| if Zi(S) >  = |P|.

Practice problems

Here are some problems to put into practice this algorithm: