Construction

int[] zFunction(String s) {
  int n = s.length();
  int[] z = new int[n];
  int l = 0, r = 0;
  for (int i = 1; i < n; i++) {
    if (i < r) z[i] = Math.min(r - i, z[i - l]);
    while (i + z[i] < n && s.charAt(z[i]) == s.charAt(i + z[i])) z[i]++;
    if (i + z[i] > r) { l = i; r = i + z[i]; }
  }
  return z;
}
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Pattern matching

Compute Z on P + '$' + T. Wherever Z[i] == |P|, pattern occurs at (i - |P| - 1) in T.

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vs KMP

Both O(N+M). Z simpler to reason about. KMP more memory-efficient during matching.

Applications

String matching. Longest common prefix queries. Palindrome checks.