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This post contains my notes and implementation of the z-algorithm.

Similar to the Bad Character Rule post this post is inspired by Gusfield's book.

The code can be found on github.

The z-algorithm finds in linear time if a pattern is a substring of text.

It starts by creating the string s following the recipe pattern$text. The character $ cannot exist in pattern or text. The choice of $ has a few interesting implications as we'll see later.

After s is built, the algorithm creates an array z with the same length of s. The index r[i], for i > 0, is the length of the longest substring starting at i that is also a prefix of s.

For example

i 0123456789
z 0102104101

After z is built, the algorithm iterates the array and if the expression r[i] == pattern.length yields true an instance of pattern has be found as a substring of text.

The array z can be built in linear time using the concept of a z-box. A z-box is a substring that is a prefix of the string. For example

    | |

The substring abc defines a z-box starting at l = 4 and ending at z = 6.

The trick to calculate the r-array in linear time is to rely in the z-box calculated during the previous iteration, i - 1, when calculating r[i]. We use the variables l and r to define the boundaries of the z-box.

There are two base cases the z-algorithm has to handle when calculating r[i].

1. If i > r, that means that i is outside the previous z-box and the algorithm compares the characters against the prefix of s. Then, if z[i] > 0 a new z-box is found and l and r are updated accordingly.

2. If i <= r, that means that i is inside of the previous z-box and there are two cases to be handled based on the value of z[i - l] and beta = r - i + 1. If z[i - l] < beta we've found a new z-box. Otherwise, we have a new z-box that starts at i but that might be larger than the current value of r.

On a first read that sounds complicated but the idea is simple. The expression i - l denotes the position of the character s(i) in the prefix of the string. The expression that defines beta is r - i + 1. The beta variable is the length defined from the index i up to the end of the z-box.

So, z[i - l] < beta indicates that no comparisons are necessary to calculate z[i] and we can define z[i] = z[i - l]. On the other hand, if z[i - l] >= beta, z[i] is at least the same length of beta.

Before we implement the function that calculates the z-array there's one more problem to tackle. The algorithm requires a character that does not appear in text or pattern. Which character should we use?

Java strings are represented in UTF-16 which guarantees that the values between U+D800 and U+DFFF are reserved code points and will never be assigned a character. A value in this range is a good candidate to be used as the separator $ in S. The problem is that this is not future proof. For example, the string representation could be updated to use UTF-8 instead of UTF-16 in a future JDK release where the range U+D800-U+DFFF is not reserved.

One alternative is to use an int instead of chars and pick a separator that is greater than 16 bits. This approach works albeit with greater memory usage.

If instead of working with chars we're interested in code points and returning code point indexes using ints is the alternative to follow.

The range of legal code points is U+0000 to U+10FFFF and according to the Character java class the lower 21 bits of an int are used to represent Unicode code points and the most significant 11 bits must be zero. In this case, the value FFE0 << 16 is a good candidate as a separator.

In our implementation we avoid having to pick an unique character identifier by calculating the z-array in two steps. We calculate the z-array of s in two steps. We first calculate the z-array slice that corresponds to the pattern followed by the remaining values of the z-array that represents the text. The position of the separator character is left untouched and carries the value zero.

The implementation follows

public class ZAlgorithm {

    private static void calculateZ(int[] z, char[] s, int ini, int end) {
        int l = 0; // z-box left
        int r = 0; // z-box right

        for (int i = ini; i <= end; i++) {
            if (i > r) { // try to find a new box z-box
                l = r = i;
                while (r <= end && s[r] == s[r - l]) r++;
                z[i] = r - l;
            } else { // inside a z-box
                int k = i - l;
                if (z[k] < (r - i + 1)) {
                    z[i] = z[k];
                } else {
                    l = i; // calculate new z-box
                    while (r <= end && s[r] == s[r - l]) r++;
                    z[i] = r - l;

    private static int[] zarray(String pattern, String text) {
        int plen = pattern.length();
        int tlen = text.length();
        int slen = plen + tlen + 1;
        char[] s = new char[slen];
        pattern.getChars(0, plen, s, 0);
        text.getChars(0, tlen, s, plen + 1);
        int[] r = new int[slen];
        calculateZ(r, s, 1, plen - 1); // pattern
        calculateZ(r, s, plen + 1, slen - 1); // text
        return r;

    public static List issubstring(String pattern, String text) {
        List indices = new ArrayList<>();
        if (pattern.length() > text.length()) return indices;
        int[] z = zarray(pattern, text);
        for (int i = pattern.length() + 1; i < z.length; i++)
            if (z[i] == pattern.length())
                indices.add(i - pattern.length() - 1);

        return indices;

As a curiosity, according to Gusfield the z-algorithm was first introduced in An O(n log n) Algorithm for Finding All Repetitions in a String.

EDIT: 2021-01-09

I've added an optimized version of the z-algorithm ZAlgorithmV2. It calculates the z-array only for the pattern while also avoiding unnecessary memory allocations.

EDIT: 2021-01-22

I've updated the code to return the indices of instances of pattern found in text and fixed a few broken links.

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