kmp_algorithm (string matching)
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The document discusses the construction of the longest prefix suffix (LPS) array for various patterns and strings, illustrating calculations for common prefixes and suffixes. An example with the pattern 'aababaab' demonstrates how to derive the LPS values for each character in the pattern. The process involves matching prefixes and suffixes, updating indices, and handling mismatches based on previously computed LPS values.
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0
lps[0] initialized to zero](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-8-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0
len=0, i=1
pat[i] != pat[len]
Since len=0, lps[i] = 0
i++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-9-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1
len=0, i=2
pat[i] == pat[len]
lps[i] = len + 1
i++, len++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-10-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2
len=1, i=3
pat[i] == pat[len]
lps[i] = len + 1
i++, len++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-11-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3
len=2, i=4
pat[i] == pat[len]
lps[i] = len + 1
i++, len++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-12-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3
len=3, i=5
pat[i] != pat[len]
len=lps[len-1]](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-13-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3
len=1, i=5
pat[i] != pat[len]
len=lps[len-1]](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-14-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3 0
len=0, i=5
pat[i] != pat[len]
Since len=0, lps[i]=0
i++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-15-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3 0 1
len=0, i=6
pat[i] == pat[len]
lps[i]=len+1
len++, i++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-16-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3 0 1 2
len=1, i=7
pat[i] == pat[len]
lps[i]=len+1
len++, i++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-17-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3 0 1 2 3
len=2, i=8
pat[i] == pat[len]
lps[i]=len+1
len++, i++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-18-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3 0 1 2 3 4
len=3, i=9
pat[i] == pat[len]
lps[i]=len+1
len++, i++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-19-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3 0 1 2 3 4 5
len=4, i=10
pat[i] == pat[len]
lps[i]=len+1
len++, i++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-20-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3 0 1 2 3 4 5 6
len=5, i=11
pat[i] == pat[len]
lps[i]=len+1
len++, i++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-21-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3 0 1 2 3 4 5 6 7
len=6, i=12
pat[i] == pat[len]
lps[i]=len+1
len++, i++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-22-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3 0 1 2 3 4 5 6 7 8
len=7, i=13
pat[i] == pat[len]
lps[i]=len+1
len++, i++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-23-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3 0 1 2 3 4 5 6 7 8 9
len=8, i=14
pat[i] == pat[len]
lps[i]=len+1
len++, i++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-24-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3 0 1 2 3 4 5 6 7 8 9 10
len=9, i=15
pat[i] == pat[len]
lps[i]=len+1
len++, i++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-25-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 11
len=10, i=16
pat[i] == pat[len]
lps[i]=len+1
len++, i++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-26-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 11
len=11, i=17
pat[i] != pat[len]
len=lps[len-1]](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-27-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 11
len=5, i=17
pat[i] != pat[len]
len=lps[len-1]](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-28-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Pattern a c a c a b a c a c a b a c a c a c
LPS 0 0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 11 4
len=3, i=17
pat[i] == pat[len]
lps[i]=len+1
len++, i++](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-29-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11
String a b x a b c a b c a b y
Index 0 1 2 3 4 5
Pattern a b c a b y
LPS Array 0 0 0 1 2 0
Now, since Pattern[j] doesn’t match with String[i], j=LPS[j-1], start=i-LPS[j-1].
i
j
start](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-33-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11
String a b x a b c a b c a b y
Index 0 1 2 3 4 5
Pattern a b c a b y
LPS Array 0 0 0 1 2 0
i
j
Pattern[j] is still not equal to String[i].And since j=0, we increment ‘i’ and ‘start’ this time.
start](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-34-320.jpg)
![Index 0 1 2 3 4 5 6 7 8 9 10 11
String a b x a b c a b c a b y
Index 0 1 2 3 4 5
Pattern a b c a b y
LPS Array 0 0 0 1 2 0
i
j
Again String[i]!=Pattern[j].Therefore j=LPS[j-1]. start=i-LPS[j-1].
start](https://image.slidesharecdn.com/daa71-to-75kmpalgorithm-170530171443/85/kmp_algorithm-string-matching-36-320.jpg)
kmp_algorithm (string matching)
- 4. length of the longest proper suffix which is also the proper prefix for a substring from zero to ‘i’.
- 5. • Proper Prefixes of “aababaab” : “a” , “aa” , “aab” , “aaba” , “aabab” , “aababa” and “aababaa”. • Proper Suffixes of “aababaab” : “b” , “ab” , “aab” , “baab” , “abaab”, “babaab” and “ababaab”. • There is only one common suffix and prefix : “aab”. Length is 3.
- 6. Index 0 1 2 3 4 5 Pattern a b c a b y LPS Array 0 0 0 1 2 0 Index 0 1 2 3 4 5 6 7 8 Pattern a a b a a b a a a LPS Array 0 1 0 1 2 3 4 5 2
- 8. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 lps[0] initialized to zero
- 9. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 len=0, i=1 pat[i] != pat[len] Since len=0, lps[i] = 0 i++
- 10. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 len=0, i=2 pat[i] == pat[len] lps[i] = len + 1 i++, len++
- 11. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 len=1, i=3 pat[i] == pat[len] lps[i] = len + 1 i++, len++
- 12. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 len=2, i=4 pat[i] == pat[len] lps[i] = len + 1 i++, len++
- 13. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 len=3, i=5 pat[i] != pat[len] len=lps[len-1]
- 14. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 len=1, i=5 pat[i] != pat[len] len=lps[len-1]
- 15. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 0 len=0, i=5 pat[i] != pat[len] Since len=0, lps[i]=0 i++
- 16. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 0 1 len=0, i=6 pat[i] == pat[len] lps[i]=len+1 len++, i++
- 17. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 0 1 2 len=1, i=7 pat[i] == pat[len] lps[i]=len+1 len++, i++
- 18. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 0 1 2 3 len=2, i=8 pat[i] == pat[len] lps[i]=len+1 len++, i++
- 19. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 0 1 2 3 4 len=3, i=9 pat[i] == pat[len] lps[i]=len+1 len++, i++
- 20. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 0 1 2 3 4 5 len=4, i=10 pat[i] == pat[len] lps[i]=len+1 len++, i++
- 21. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 0 1 2 3 4 5 6 len=5, i=11 pat[i] == pat[len] lps[i]=len+1 len++, i++
- 22. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 0 1 2 3 4 5 6 7 len=6, i=12 pat[i] == pat[len] lps[i]=len+1 len++, i++
- 23. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 0 1 2 3 4 5 6 7 8 len=7, i=13 pat[i] == pat[len] lps[i]=len+1 len++, i++
- 24. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 0 1 2 3 4 5 6 7 8 9 len=8, i=14 pat[i] == pat[len] lps[i]=len+1 len++, i++
- 25. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 len=9, i=15 pat[i] == pat[len] lps[i]=len+1 len++, i++
- 26. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 11 len=10, i=16 pat[i] == pat[len] lps[i]=len+1 len++, i++
- 27. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 11 len=11, i=17 pat[i] != pat[len] len=lps[len-1]
- 28. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 11 len=5, i=17 pat[i] != pat[len] len=lps[len-1]
- 29. Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Pattern a c a c a b a c a c a b a c a c a c LPS 0 0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 11 4 len=3, i=17 pat[i] == pat[len] lps[i]=len+1 len++, i++
- 31. Index 0 1 2 3 4 5 6 7 8 9 10 11 String a b x a b c a b c a b y Index 0 1 2 3 4 5 Pattern a b c a b y LPS Array 0 0 0 1 2 0 i j start
- 32. Index 0 1 2 3 4 5 6 7 8 9 10 11 String a b x a b c a b c a b y Index 0 1 2 3 4 5 Pattern a b c a b y LPS Array 0 0 0 1 2 0 i j start
- 33. Index 0 1 2 3 4 5 6 7 8 9 10 11 String a b x a b c a b c a b y Index 0 1 2 3 4 5 Pattern a b c a b y LPS Array 0 0 0 1 2 0 Now, since Pattern[j] doesn’t match with String[i], j=LPS[j-1], start=i-LPS[j-1]. i j start
- 34. Index 0 1 2 3 4 5 6 7 8 9 10 11 String a b x a b c a b c a b y Index 0 1 2 3 4 5 Pattern a b c a b y LPS Array 0 0 0 1 2 0 i j Pattern[j] is still not equal to String[i].And since j=0, we increment ‘i’ and ‘start’ this time. start
- 35. Index 0 1 2 3 4 5 6 7 8 9 10 11 String a b x a b c a b c a b y Index 0 1 2 3 4 5 Pattern a b c a b y LPS Array 0 0 0 1 2 0 i j start
- 36. Index 0 1 2 3 4 5 6 7 8 9 10 11 String a b x a b c a b c a b y Index 0 1 2 3 4 5 Pattern a b c a b y LPS Array 0 0 0 1 2 0 i j Again String[i]!=Pattern[j].Therefore j=LPS[j-1]. start=i-LPS[j-1]. start
- 37. Index 0 1 2 3 4 5 6 7 8 9 10 11 String a b x a b c a b c a b y Index 0 1 2 3 4 5 Pattern a b c a b y LPS Array 0 0 0 1 2 0 i j start
- 38. Index 0 1 2 3 4 5 6 7 8 9 10 11 String a b x a b c a b c a b y Index 0 1 2 3 4 5 Pattern a b c a b y LPS Array 0 0 0 1 2 0 i j start We reached the end of the pattern. Hence the pattern is found at index ‘start’.