This is another post containing my notes and implementation of the z-algorithm inspired by Gusfield’s book.

The code can be found here.

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
aabaaxaaba
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:

``````01234567
abcxabcy
| |
---
z-box
``````

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’re only interested if `pattern` exists or not in `text`. Our constraint is more relaxed and we can have an implementation that handles chars and the only gotcha is to calculate the `z` array in two steps. First we calculate `z` that corresponds to the `pattern` part, then we skip the separator position and calculate the `z` array slice representing the `text` portion.

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) {
l = r = i;
while (r <= end && s[r] == s[r - l]) r++;
z[i] = r - l;
r--;
} else {
int k = i - l;
if (z[k] < (r - i + 1)) {
z[i] = z[k];
} else {
l = i;
while (r <= end && s[r] == s[r - l]) r++;
z[i] = r - l;
r--;
}
}
}
}

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 - 1, s, 0);
text.getChars(0, tlen - 1, 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 boolean issubstring(String pattern, String text) {
if (pattern.length() > text.length()) return false;
int[] z = zarray(pattern, text);
for (int i = pattern.length() + 1; i < z.length; i++)
if (z[i] == pattern.length())
return true;

return false;
}

}
``````

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.