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KMP Algorithm for Pattern Searching

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  • Difficulty Level : Hard
  • Last Updated : 30 Sep, 2022
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Given a text txt[0..n-1] and a pattern pat[0..m-1], write a function search(char pat[], char txt[]) that prints all occurrences of pat[] in txt[]. You may assume that n > m

Examples:

Input:  txt[] = "THIS IS A TEST TEXT"
        pat[] = "TEST"
Output: Pattern found at index 10

Input:  txt[] =  "AABAACAADAABAABA"
        pat[] =  "AABA"
Output: Pattern found at index 0
        Pattern found at index 9
        Pattern found at index 12
        

 

Pattern searching is an important problem in computer science. When we do search for a string in notepad/word file or browser or database, pattern searching algorithms are used to show the search results. We have discussed Naive pattern searching algorithm in the previous post. The worst case complexity of the Naive algorithm is O(m(n-m+1)). The time complexity of KMP algorithm is O(n) in the worst case. KMP (Knuth Morris Pratt) Pattern Searching The Naive pattern searching algorithm doesn’t work well in cases where we see many matching characters followed by a mismatching character. Following are some examples.

   txt[] = "AAAAAAAAAAAAAAAAAB"
   pat[] = "AAAAB"

   txt[] = "ABABABCABABABCABABABC"
   pat[] =  "ABABAC" (not a worst case, but a bad case for Naive)

The KMP matching algorithm uses degenerating property (pattern having same sub-patterns appearing more than once in the pattern) of the pattern and improves the worst case complexity to O(n). The basic idea behind KMP’s algorithm is: whenever we detect a mismatch (after some matches), we already know some of the characters in the text of the next window. We take advantage of this information to avoid matching the characters that we know will anyway match. Let us consider below example to understand this.

Matching Overview
txt = "AAAAABAAABA" 
pat = "AAAA"

We compare first window of txt with pat
txt = "AAAAABAAABA" 
pat = "AAAA"  [Initial position]
We find a match. This is same as Naive String Matching.

In the next step, we compare next window of txt with pat.
txt = "AAAAABAAABA" 
pat =  "AAAA" [Pattern shifted one position]
This is where KMP does optimization over Naive. In this 
second window, we only compare fourth A of pattern
with fourth character of current window of text to decide 
whether current window matches or not. Since we know 
first three characters will anyway match, we skipped 
matching first three characters. 

Need of Preprocessing?
An important question arises from the above explanation, 
how to know how many characters to be skipped. To know this, 
we pre-process pattern and prepare an integer array 
lps[] that tells us the count of characters to be skipped. 

Preprocessing Overview:

  • KMP algorithm preprocesses pat[] and constructs an auxiliary lps[] of size m (same as size of pattern) which is used to skip characters while matching.
  • name lps indicates longest proper prefix which is also suffix.. A proper prefix is prefix with whole string not allowed. For example, prefixes of “ABC” are “”, “A”, “AB” and “ABC”. Proper prefixes are “”, “A” and “AB”. Suffixes of the string are “”, “C”, “BC” and “ABC”.
  • We search for lps in sub-patterns. More clearly we focus on sub-strings of patterns that are either prefix and suffix.
  • For each sub-pattern pat[0..i] where i = 0 to m-1, lps[i] stores length of the maximum matching proper prefix which is also a suffix of the sub-pattern pat[0..i].
   lps[i] = the longest proper prefix of pat[0..i] 
              which is also a suffix of pat[0..i]. 

Note : lps[i] could also be defined as longest prefix which is also proper suffix. We need to use properly at one place to make sure that the whole substring is not considered.

Examples of lps[] construction:
For the pattern “AAAA”, 
lps[] is [0, 1, 2, 3]

For the pattern “ABCDE”, 
lps[] is [0, 0, 0, 0, 0]

For the pattern “AABAACAABAA”, 
lps[] is [0, 1, 0, 1, 2, 0, 1, 2, 3, 4, 5]

For the pattern “AAACAAAAAC”, 
lps[] is [0, 1, 2, 0, 1, 2, 3, 3, 3, 4] 

For the pattern “AAABAAA”, 
lps[] is [0, 1, 2, 0, 1, 2, 3]

Searching Algorithm:
Unlike Naive algorithm, where we slide the pattern by one and compare all characters at each shift, we use a value from lps[] to decide the next characters to be matched. The idea is to not match a character that we know will anyway match.

How to use lps[] to decide next positions (or to know a number of characters to be skipped)?

  • We start comparison of pat[j] with j = 0 with characters of current window of text.
  • We keep matching characters txt[i] and pat[j] and keep incrementing i and j while pat[j] and txt[i] keep matching.
  • When we see a mismatch
    • We know that characters pat[0..j-1] match with txt[i-j…i-1] (Note that j starts with 0 and increment it only when there is a match).
    • We also know (from above definition) that lps[j-1] is count of characters of pat[0…j-1] that are both proper prefix and suffix.
    • From above two points, we can conclude that we do not need to match these lps[j-1] characters with txt[i-j…i-1] because we know that these characters will anyway match. Let us consider above example to understand this.
txt[] = "AAAAABAAABA" 
pat[] = "AAAA"
lps[] = {0, 1, 2, 3} 

i = 0, j = 0
txt[] = "AAAAABAAABA" 
pat[] = "AAAA"
txt[i] and pat[j] match, do i++, j++

i = 1, j = 1
txt[] = "AAAAABAAABA" 
pat[] = "AAAA"
txt[i] and pat[j] match, do i++, j++

i = 2, j = 2
txt[] = "AAAAABAAABA" 
pat[] = "AAAA"
pat[i] and pat[j] match, do i++, j++

i = 3, j = 3
txt[] = "AAAAABAAABA" 
pat[] = "AAAA"
txt[i] and pat[j] match, do i++, j++

i = 4, j = 4
Since j == M, print pattern found and reset j,
j = lps[j-1] = lps[3] = 3

Here unlike Naive algorithm, we do not match first three 
characters of this window. Value of lps[j-1] (in above 
step) gave us index of next character to match.
i = 4, j = 3
txt[] = "AAAAABAAABA" 
pat[] =  "AAAA"
txt[i] and pat[j] match, do i++, j++

i = 5, j = 4
Since j == M, print pattern found and reset j,
j = lps[j-1] = lps[3] = 3

Again unlike Naive algorithm, we do not match first three 
characters of this window. Value of lps[j-1] (in above 
step) gave us index of next character to match.
i = 5, j = 3
txt[] = "AAAAABAAABA" 
pat[] =   "AAAA"
txt[i] and pat[j] do NOT match and j > 0, change only j
j = lps[j-1] = lps[2] = 2

i = 5, j = 2
txt[] = "AAAAABAAABA" 
pat[] =    "AAAA"
txt[i] and pat[j] do NOT match and j > 0, change only j
j = lps[j-1] = lps[1] = 1 

i = 5, j = 1
txt[] = "AAAAABAAABA" 
pat[] =     "AAAA"
txt[i] and pat[j] do NOT match and j > 0, change only j
j = lps[j-1] = lps[0] = 0

i = 5, j = 0
txt[] = "AAAAABAAABA" 
pat[] =      "AAAA"
txt[i] and pat[j] do NOT match and j is 0, we do i++.

i = 6, j = 0
txt[] = "AAAAABAAABA" 
pat[] =       "AAAA"
txt[i] and pat[j] match, do i++ and j++

i = 7, j = 1
txt[] = "AAAAABAAABA" 
pat[] =       "AAAA"
txt[i] and pat[j] match, do i++ and j++

We continue this way till there are sufficient characters in the text to be compared with the characters in the pattern...

C++




// C++ program for implementation of KMP pattern searching
// algorithm
#include <bits/stdc++.h>
 
void computeLPSArray(char* pat, int M, int* lps);
 
// Prints occurrences of txt[] in pat[]
void KMPSearch(char* pat, char* txt)
{
    int M = strlen(pat);
    int N = strlen(txt);
 
    // create lps[] that will hold the longest prefix suffix
    // values for pattern
    int lps[M];
 
    // Preprocess the pattern (calculate lps[] array)
    computeLPSArray(pat, M, lps);
 
    int i = 0; // index for txt[]
    int j = 0; // index for pat[]
    while ((N - i) >= (M - j)) {
        if (pat[j] == txt[i]) {
            j++;
            i++;
        }
 
        if (j == M) {
            printf("Found pattern at index %d ", i - j);
            j = lps[j - 1];
        }
 
        // mismatch after j matches
        else if (i < N && pat[j] != txt[i]) {
            // Do not match lps[0..lps[j-1]] characters,
            // they will match anyway
            if (j != 0)
                j = lps[j - 1];
            else
                i = i + 1;
        }
    }
}
 
// Fills lps[] for given pattern pat[0..M-1]
void computeLPSArray(char* pat, int M, int* lps)
{
    // length of the previous longest prefix suffix
    int len = 0;
 
    lps[0] = 0; // lps[0] is always 0
 
    // the loop calculates lps[i] for i = 1 to M-1
    int i = 1;
    while (i < M) {
        if (pat[i] == pat[len]) {
            len++;
            lps[i] = len;
            i++;
        }
        else // (pat[i] != pat[len])
        {
            // This is tricky. Consider the example.
            // AAACAAAA and i = 7. The idea is similar
            // to search step.
            if (len != 0) {
                len = lps[len - 1];
 
                // Also, note that we do not increment
                // i here
            }
            else // if (len == 0)
            {
                lps[i] = 0;
                i++;
            }
        }
    }
}
 
// Driver program to test above function
int main()
{
    char txt[] = "ABABDABACDABABCABAB";
    char pat[] = "ABABCABAB";
    KMPSearch(pat, txt);
    return 0;
}


Java




// JAVA program for implementation of KMP pattern
// searching algorithm
 
class KMP_String_Matching {
    void KMPSearch(String pat, String txt)
    {
        int M = pat.length();
        int N = txt.length();
 
        // create lps[] that will hold the longest
        // prefix suffix values for pattern
        int lps[] = new int[M];
        int j = 0; // index for pat[]
 
        // Preprocess the pattern (calculate lps[]
        // array)
        computeLPSArray(pat, M, lps);
 
        int i = 0; // index for txt[]
        while ((N - i) >= (M - j)) {
            if (pat.charAt(j) == txt.charAt(i)) {
                j++;
                i++;
            }
            if (j == M) {
                System.out.println("Found pattern "
                                + "at index " + (i - j));
                j = lps[j - 1];
            }
 
            // mismatch after j matches
            else if (i < N && pat.charAt(j) != txt.charAt(i)) {
                // Do not match lps[0..lps[j-1]] characters,
                // they will match anyway
                if (j != 0)
                    j = lps[j - 1];
                else
                    i = i + 1;
            }
        }
    }
 
    void computeLPSArray(String pat, int M, int lps[])
    {
        // length of the previous longest prefix suffix
        int len = 0;
        int i = 1;
        lps[0] = 0; // lps[0] is always 0
 
        // the loop calculates lps[i] for i = 1 to M-1
        while (i < M) {
            if (pat.charAt(i) == pat.charAt(len)) {
                len++;
                lps[i] = len;
                i++;
            }
            else // (pat[i] != pat[len])
            {
                // This is tricky. Consider the example.
                // AAACAAAA and i = 7. The idea is similar
                // to search step.
                if (len != 0) {
                    len = lps[len - 1];
 
                    // Also, note that we do not increment
                    // i here
                }
                else // if (len == 0)
                {
                    lps[i] = len;
                    i++;
                }
            }
        }
    }
 
    // Driver program to test above function
    public static void main(String args[])
    {
        String txt = "ABABDABACDABABCABAB";
        String pat = "ABABCABAB";
        new KMP_String_Matching().KMPSearch(pat, txt);
    }
}
// This code has been contributed by Amit Khandelwal.


Python




# Python program for KMP Algorithm
def KMPSearch(pat, txt):
    M = len(pat)
    N = len(txt)
 
    # create lps[] that will hold the longest prefix suffix
    # values for pattern
    lps = [0]*M
    j = 0 # index for pat[]
 
    # Preprocess the pattern (calculate lps[] array)
    computeLPSArray(pat, M, lps)
 
    i = 0 # index for txt[]
    while (N - i) >= (M - j):
        if pat[j] == txt[i]:
            i += 1
            j += 1
 
        if j == M:
            print ("Found pattern at index " + str(i-j))
            j = lps[j-1]
 
        # mismatch after j matches
        elif i < N and pat[j] != txt[i]:
            # Do not match lps[0..lps[j-1]] characters,
            # they will match anyway
            if j != 0:
                j = lps[j-1]
            else:
                i += 1
 
def computeLPSArray(pat, M, lps):
    len = 0 # length of the previous longest prefix suffix
 
    lps[0] # lps[0] is always 0
    i = 1
 
    # the loop calculates lps[i] for i = 1 to M-1
    while i < M:
        if pat[i]== pat[len]:
            len += 1
            lps[i] = len
            i += 1
        else:
            # This is tricky. Consider the example.
            # AAACAAAA and i = 7. The idea is similar
            # to search step.
            if len != 0:
                len = lps[len-1]
 
                # Also, note that we do not increment i here
            else:
                lps[i] = 0
                i += 1
 
txt = "ABABDABACDABABCABAB"
pat = "ABABCABAB"
KMPSearch(pat, txt)
 
# This code is contributed by Bhavya Jain


C#




// C# program for implementation of KMP pattern
// searching algorithm
using System;
 
class GFG {
 
    void KMPSearch(string pat, string txt)
    {
        int M = pat.Length;
        int N = txt.Length;
 
        // create lps[] that will hold the longest
        // prefix suffix values for pattern
        int[] lps = new int[M];
        int j = 0; // index for pat[]
 
        // Preprocess the pattern (calculate lps[]
        // array)
        computeLPSArray(pat, M, lps);
 
        int i = 0; // index for txt[]
        while ((N - i) >= (M - j)) {
            if (pat[j] == txt[i]) {
                j++;
                i++;
            }
            if (j == M) {
                Console.Write("Found pattern "
                            + "at index " + (i - j));
                j = lps[j - 1];
            }
 
            // mismatch after j matches
            else if (i < N && pat[j] != txt[i]) {
                // Do not match lps[0..lps[j-1]] characters,
                // they will match anyway
                if (j != 0)
                    j = lps[j - 1];
                else
                    i = i + 1;
            }
        }
    }
 
    void computeLPSArray(string pat, int M, int[] lps)
    {
        // length of the previous longest prefix suffix
        int len = 0;
        int i = 1;
        lps[0] = 0; // lps[0] is always 0
 
        // the loop calculates lps[i] for i = 1 to M-1
        while (i < M) {
            if (pat[i] == pat[len]) {
                len++;
                lps[i] = len;
                i++;
            }
            else // (pat[i] != pat[len])
            {
                // This is tricky. Consider the example.
                // AAACAAAA and i = 7. The idea is similar
                // to search step.
                if (len != 0) {
                    len = lps[len - 1];
 
                    // Also, note that we do not increment
                    // i here
                }
                else // if (len == 0)
                {
                    lps[i] = len;
                    i++;
                }
            }
        }
    }
 
    // Driver program to test above function
    public static void Main()
    {
        string txt = "ABABDABACDABABCABAB";
        string pat = "ABABCABAB";
        new GFG().KMPSearch(pat, txt);
    }
}
 
// This code has been contributed by Amit Khandelwal.


Javascript




<script>
    //Javascript program for implementation of KMP pattern
    // searching algorithm
     
    function computeLPSArray(pat, M, lps)
    {
        // length of the previous longest prefix suffix
        var len = 0;
        var i = 1;
        lps[0] = 0; // lps[0] is always 0
     
        // the loop calculates lps[i] for i = 1 to M-1
        while (i < M) {
            if (pat.charAt(i) == pat.charAt(len)) {
                len++;
                lps[i] = len;
                i++;
            }
            else // (pat[i] != pat[len])
            {
                // This is tricky. Consider the example.
                // AAACAAAA and i = 7. The idea is similar
                // to search step.
                if (len != 0) {
                    len = lps[len - 1];
     
                    // Also, note that we do not increment
                    // i here
                }
                else // if (len == 0)
                {
                    lps[i] = len;
                    i++;
                }
            }
        }
    }
     
    function KMPSearch(pat,txt)
    {
        var M = pat.length;
        var N = txt.length;
     
        // create lps[] that will hold the longest
        // prefix suffix values for pattern
        var lps = [];
        var j = 0; // index for pat[]
     
        // Preprocess the pattern (calculate lps[]
        // array)
        computeLPSArray(pat, M, lps);
     
        var i = 0; // index for txt[]
        while ((N - i) >= (M - j)) {
            if (pat.charAt(j) == txt.charAt(i)) {
                j++;
                i++;
            }
            if (j == M) {
                document.write("Found pattern " + "at index " + (i - j) + "\n");
                j = lps[j - 1];
            }
     
            // mismatch after j matches
            else if (i < N && pat.charAt(j) != txt.charAt(i)) {
                // Do not match lps[0..lps[j-1]] characters,
                // they will match anyway
                if (j != 0)
                    j = lps[j - 1];
                else
                    i = i + 1;
            }
        }
    }
     
     
    var txt = "ABABDABACDABABCABAB";
    var pat = "ABABCABAB";
    KMPSearch(pat, txt);
    //This code is contributed by shruti456rawal
</script>


PHP




<?php
// PHP program for implementation of KMP pattern searching
// algorithm
 
 
// Prints occurrences of txt[] in pat[]
function KMPSearch($pat, $txt)
{
    $M = strlen($pat);
    $N = strlen($txt);
 
    // create lps[] that will hold the longest prefix suffix
    // values for pattern
    $lps=array_fill(0,$M,0);
 
    // Preprocess the pattern (calculate lps[] array)
    computeLPSArray($pat, $M, $lps);
 
    $i = 0; // index for txt[]
    $j = 0; // index for pat[]
    while (($N - $i) >= ($M - $j)) {
        if ($pat[$j] == $txt[$i]) {
            $j++;
            $i++;
        }
 
        if ($j == $M) {
            printf("Found pattern at index ".($i - $j));
            $j = $lps[$j - 1];
        }
 
        // mismatch after j matches
        else if ($i < $N && $pat[$j] != $txt[$i]) {
            // Do not match lps[0..lps[j-1]] characters,
            // they will match anyway
            if ($j != 0)
                $j = $lps[$j - 1];
            else
                $i = $i + 1;
        }
    }
}
 
// Fills lps[] for given pattern pat[0..M-1]
function computeLPSArray($pat, $M, &$lps)
{
    // length of the previous longest prefix suffix
    $len = 0;
 
    $lps[0] = 0; // lps[0] is always 0
 
    // the loop calculates lps[i] for i = 1 to M-1
    $i = 1;
    while ($i < $M) {
        if ($pat[$i] == $pat[$len]) {
            $len++;
            $lps[$i] = $len;
            $i++;
        }
        else // (pat[i] != pat[len])
        {
            // This is tricky. Consider the example.
            // AAACAAAA and i = 7. The idea is similar
            // to search step.
            if ($len != 0) {
                $len = $lps[$len - 1];
 
                // Also, note that we do not increment
                // i here
            }
            else // if (len == 0)
            {
                $lps[$i] = 0;
                $i++;
            }
        }
    }
}
 
// Driver program to test above function
 
    $txt = "ABABDABACDABABCABAB";
    $pat = "ABABCABAB";
    KMPSearch($pat, $txt);
     
// This code is contributed by chandan_jnu
?>


Output

Found pattern at index 10 

Preprocessing Algorithm:
In the preprocessing part, we calculate values in lps[]. To do that, we keep track of the length of the longest prefix suffix value (we use len variable for this purpose) for the previous index. We initialize lps[0] and len as 0. If pat[len] and pat[i] match, we increment len by 1 and assign the incremented value to lps[i]. If pat[i] and pat[len] do not match and len is not 0, we update len to lps[len-1]. See computeLPSArray () in the below code for details.

Illustration of preprocessing (or construction of lps[])

pat[] = "AAACAAAA"

len = 0, i  = 0.
lps[0] is always 0, we move 
to i = 1

len = 0, i  = 1.
Since pat[len] and pat[i] match, do len++, 
store it in lps[i] and do i++.
len = 1, lps[1] = 1, i = 2

len = 1, i  = 2.
Since pat[len] and pat[i] match, do len++, 
store it in lps[i] and do i++.
len = 2, lps[2] = 2, i = 3

len = 2, i  = 3.
Since pat[len] and pat[i] do not match, and len > 0, 
set len = lps[len-1] = lps[1] = 1

len = 1, i  = 3.
Since pat[len] and pat[i] do not match and len > 0, 
len = lps[len-1] = lps[0] = 0

len = 0, i  = 3.
Since pat[len] and pat[i] do not match and len = 0, 
Set lps[3] = 0 and i = 4.
We know that characters pat
len = 0, i  = 4.
Since pat[len] and pat[i] match, do len++, 
store it in lps[i] and do i++.
len = 1, lps[4] = 1, i = 5

len = 1, i  = 5.
Since pat[len] and pat[i] match, do len++, 
store it in lps[i] and do i++.
len = 2, lps[5] = 2, i = 6

len = 2, i  = 6.
Since pat[len] and pat[i] match, do len++, 
store it in lps[i] and do i++.
len = 3, lps[6] = 3, i = 7

len = 3, i  = 7.
Since pat[len] and pat[i] do not match and len > 0,
set len = lps[len-1] = lps[2] = 2

len = 2, i  = 7.
Since pat[len] and pat[i] match, do len++, 
store it in lps[i] and do i++.
len = 3, lps[7] = 3, i = 8

We stop here as we have constructed the whole lps[].

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