Prefix function - Knuth-Morris-Pratt Algorithm

Malay Gain
Last Updated: May 13, 2022


KMP is the most popular substring search algorithm in linear time. This algorithm is used for finding linear time solutions of various string-related problems such as searching substring in a string, the number of unique substrings in a string, pattern matching problem, etc.


The main component of the KMP algorithm is the prefix function. It is defined as for a string s of length n, π[i] is the length of the longest proper prefix of the substring s[0…i], which is also a suffix of this substring where the proper prefix of a string is a prefix that is not equal to the string itself. π[i] is the ith element of an n-length array π. By definition, π[0]=0.


π[i]= max



k : s[0…k−1]=s[i−(k−1)…i],

where s[0…k−1] is the proper prefix and s[i−(k−1)…i] is the suffix of length k of the string s.


e.g  for string "ababaca", prefix function array π is [0,0,1,2,3,0,1]




Brute force approach

vector<int> prefix_function(string s) 
    int n = (int)s.length();
    vector<int> pi(n);
    for (int i = 0; i < n; i++)
        for (int k = 0; k <= i; k++)
         if (s.substr(0, k) == s.substr(i-k+1, k))
             pi[i] = k;
    return pi;


It takes O(n3) in the worst case.

Optimized approach:

  • At first, compute the prefix values π[i] in a loop by iterating from i=1 to i=n−1 (π[0] is assigned with 0).
  • To calculate the current value π[i], set the variable j denoting the length of the best suffix for i−1. Initially, j=π[i−1].
  • Test if the suffix of substring s[0....i] of length j+1 is also a prefix of the substring s[0....i] by comparing s[j] and s[i]. If they are equal, we assign π[i]=j+1 (the values of the prefix function can only increase by at most one); otherwise, reduce j to π[j−1] ( greatest k<j, for which the prefix property holds is π[j−1]) and repeat this step.
  • If length j=0 is reached and still don't have a match, then assign π[i]=0 and go to the next index i+1



vector<int> prefix_function(string s) 
    int n = (int)s.length();
    vector<int> pi(n);
    for (int i = 1; i < n; i++) 
        int j = pi[i-1];
        while (j > 0 && s[i] != s[j])
            j = pi[j-1];
        if (s[i] == s[j])
        pi[i] = j;
    return pi;


Complexity analysis

The KMP algorithm takes O(n) time in the worst case and space complexity is also O(n) for using an extra array of length n.



Searching a substring in a string

Searching a pattern P in string S using the ∏ function is just like making the ∏ function. If we get any difference in S and P, we can jump on a character in S with the help of ∏ function because we already equaled the suffix and prefix. Go through the below example for understanding how prefix function is used for pattern matching.





using namespace std;

void findPrefix(string P, int m, int prefArray[]) {
  int length = 0;
  prefArray[0] = 0;     //first place is always 0 as no prefix

  for(int i = 1; i<m; i++) {
      if(P[i] == P[length]) {
        prefArray[i] = length;    
      }else {
        if(length != 0) {
            length = prefArray[length - 1];
            i--;     //decrease i to avoid effect of increasing after iteration
            prefArray[i] = 0;

void KMPSearch(string S, string P, int *locArray, int &loc) {
  int n, m, i = 0, j = 0;
  n = S.size();
  m = P.size();
  int prefixArray[m];    //prefix(?) array as same size of P 
  findPrefix(P, m, prefixArray);
  loc = 0;

  while(i < n) {
      if(S[i] == P[j]) {

      if(j == m) {
        locArray[loc] = i-j;      //item found at i-j position.
        j = prefixArray[j-1];    //get the prefix length from array
      }else if(i < n && P[j] != S[i]) {
        if(j != 0)
            j = prefixArray[j-1];

int main() {
  string str = "aaaabaaaaabbbaaaab";
  string patt = "aaab";
  int locationArray[str.size()]; // array to store the start locations from where the pattern matched
  int index;
  KMPSearch(str, patt, locationArray, index);

  for(int i = 0; i<index; i++) {
      cout << "Pattern found at location: " <<locationArray[i] << endl;



Pattern found at location: 1
Pattern found at location: 7
Pattern found at location: 14




  1. What is the application of the KMP algorithm?
    KMP algorithm is used for searching a substring in a string, the number of unique substrings in a string, pattern matching problem.
  2. What is the alternative algorithm of the Knuth-Morris-Pratt algorithm?
    Robin-Karp algorithm is an alternative to the KMP algorithm.
  3. What is the proper prefix of a string?
    The proper prefix of a string is a substring that is not equal to the length of the string itself.


Key Takeaways

This article covered the KMP algorithm. We have learned about the algorithm to apply it for solving various string-related problems.

Now, you can practice and solve a wide range of string-related coding questions commonly asked in interviews in CodeStudio. Along with coding questions, we can also find the interview experience of scholars working in renowned product-based companies here. 

Happy learning!

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