Knuth–Morris–Pratt Algorithm for Pattern Searching (String Matching Algo)


Knuth–Morris–Pratt Algorithm for Pattern Searching

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.

Problem :
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.

Example :

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

Idea :
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.

Preprocessing Pattern String :

  • 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 patternABCDE”, lps[] is [0, 0, 0, 0, 0] For the patternAABAACAABAA”, lps[] is [0, 1, 0, 1, 2, 0, 1, 2, 3, 4, 5]

Searching Algorithm :

In this algorithm, 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+1…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 the 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.

Code Of KMP Algorithm in C++

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++; } } } }
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 (i < N) { 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; } } }

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