Tuned Boyer Moore Algorithm Raita Algorithm Horspool Algorithm
Tuned Boyer Moore Algorithm Raita Algorithm Horspool Algorithm Quick Search Algorithm Smith Algorithm Zhu-Takaoka Algorithm Adviser: R. C. T. Lee Speaker: C. W. Cheng National Chi Nan University 1
Problem Definition Input: a text string T with length n and a pattern string P with length m. Output: all occurrences of P in T. 2
Definition • Ts : the first character of a string T aligns to a pattern P. • Pl : the first character of a pattern P aligns to a string T. • Tj : the character of the jth position of a string T. • Pi : the character of the ith position of a pattern P. • Pf : the last character of a pattern P. • n : The length of T. • m : The length of P. 3
Rule 2 -2: 1 -Suffix Rule (A Special Version of Rule 2) • Consider the 1 -suffix x. We may apply Rule 2 -2 now. 4
Tuned Boyer Moore Algorithm Fast string searching , HUME A. and SUNDAY D. M. , Software - Practice & Experience 21(11), 1991, pp. 1221 -1248. 5
Introduction • simplification of the Boyer-Moore algorithm. • uses only the bad-character shift. • easy to implement. • very fast in practice • uses Rule 2 -2: 1 -Suffix Rule 6
Tuned Boyer Moore Algorithm • In this algorithm, We always focus on the last character of the window of T and try to slide the pattern to match the last character of T. 7
Tuned Boyer Moore Algorithm Rule Since Ts+m-1 ≠ Pf , we move the pattern P to right such that the largest position i in the right of Pi is equal to Ts+m. We can shift the pattern at least (m-i) positions right until Ts+m-1 = Pf. s s+m-1 T x P z x i 1 Shift P 1 z y y f z i Shift x y f P 1 z i x y f 8
Tuned Boyer Moore Preprocessing Table • In this algorithm, we construct a table as follow. Let x be a character in the alphabet. We record the position of the last x, if it exists in P, we record the position of x from the second last position of P. If x does not exist in P 1 to Pm-1, we record it as m. 9
Tuned Boyer Moore Preprocessing Table • Example： 654321 P=AGCAGAC bm. BC A C G T 1 4 2 7 10
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC tbm. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG AGCAGAC 11
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC tbm. BC A C G T 1 4 2 7 tbm. BC[A]=1, shift=1 GCGAGCAGACGTGCGAGTACG AGCAGAC 12
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC tbm. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 13
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC tbm. BC A C G T 1 4 2 7 tbm. BC[G]=2, shift=2 GCGAGCAGACGTGCGAGTACG AGCAGAC 14
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC tbm. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 15
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC tbm. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG match AGCAGAC 16
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC tbm. BC A C G T 1 4 2 7 tbm. BC[C]=4, shift=4 GCGAGCAGACGTGCGAGTACG exact match AGCAGAC 17
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC tbm. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 18
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC tbm. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG match AGCAGAC 19
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC tbm. BC A C G T 1 4 2 7 tbm. BC[C]=4, shift=4 GCGAGCAGACGTGCGAGTACG mismatch AGCAGAC 20
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC tbm. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 21
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC tbm. BC A C G T 1 4 2 7 tbm. BC[T]=7, shift=7 GCGAGCAGACGTGCGAGTACG AGCAGAC 22
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC tbm. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 23
Time complexity • preprocessing phase in O(m+ σ) time and O(σ) space complexity, σ is the number of alphabets in pattern. • searching phase in O(mn) time complexity. 24
Raita Algorithm Tuning the Boyer-Moore-Horspool string searching algorithm, T. RAITA, Software Practice & Experience, 22(10), 1994, pp. 879 -884 25
Introduction • simplification of the Boyer-Moore algorithm. • uses only the bad-character shift. • easy to implement. • very fast in practice • uses Rule 2 -2: 1 -Suffix Rule 26
Raita Algorithm • In this algorithm, first we compare the last character of the window of T with the last character of the pattern, then we compare the first character and the middle character of the window. If they match, we compare other characters from left to right. If mismatch occurs, we slide the window by the preprocessing table. 27
Raita Preprocessing Table • The preprocessing table of Raita algorithm is the same with Tuned Boyer-Moore algorithm. 28
Raita Preprocessing Table • Example： 654321 P=AGCAGAC ra. BC A C G T 1 4 2 7 29
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC ra. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG AGCAGAC 30
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC ra. BC A C G T 1 4 2 7 ra. BC[A]=1, shift=1 GCGAGCAGACGTGCGAGTACG AGCAGAC 31
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC ra. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 32
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC ra. BC A C G T 1 4 2 7 ra. BC[G]=2, shift=2 GCGAGCAGACGTGCGAGTACG AGCAGAC 33
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC ra. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 34
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC ra. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG match AGCAGAC 35
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC ra. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG match AGCAGAC 36
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC ra. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG match AGCAGAC 37
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC ra. BC A C G T 1 4 2 7 ra. BC[C]=4, shift=4 GCGAGCAGACGTGCGAGTACG exact match AGCAGAC 38
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC ra. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 39
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC ra. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG match AGCAGAC 40
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC ra. BC A C G T 1 4 2 7 ra. BC[C]=4, shift=4 GCGAGCAGACGTGCGAGTACG mismatch AGCAGAC 41
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC ra. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 42
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC ra. BC A C G T 1 4 2 7 ra. BC[T]=7, shift=7 GCGAGCAGACGTGCGAGTACG AGCAGAC 43
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC ra. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 44
Time complexity • preprocessing phase in O(m+ σ) time and O(σ) space complexity, σ is the number of alphabets in pattern. • searching phase in O(mn) time complexity. 45
Horspool Algorithm Practical fast searching in strings, R. NIGEL HORSPOOL, SOFTWAREPRACTICE AND EXPERIENCE, VOL. 10, 1980, pp. 501 -506 46
Introduction • simplification of the Boyer-Moore algorithm. • uses only the bad-character shift. • easy to implement. • very fast in practice • uses Rule 2 -2: 1 -Suffix Rule 47
Horspool Algorithm • In this algorithm, We always compare the window of T with pattern from right to left and try to slide the pattern to match the last character of T. 48
Horspool Preprocessing Table • The preprocessing table of Horspool algorithm is the same with Tuned Boyer-Moore algorithm. 49
Horspool Preprocessing Table • Example： 654321 P=AGCAGAC hp. BC A C G T 1 4 2 7 50
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG AGCAGAC 51
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 hp. BC[A]=1, shift=1 GCGAGCAGACGTGCGAGTACG AGCAGAC 52
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 53
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 hp. BC[G]=2, shift=2 GCGAGCAGACGTGCGAGTACG AGCAGAC 54
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 55
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG match AGCAGAC 56
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG match AGCAGAC 57
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG match AGCAGAC 58
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 hp. BC[C]=4, shift=4 GCGAGCAGACGTGCGAGTACG exact match AGCAGAC 59
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 60
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG match AGCAGAC 61
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 hp. BC[G]=2, shift=2 GCGAGCAGACGTGCGAGTACG mismatch AGCAGAC 62
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 63
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 hp. BC[A]=1, shift=1 GCGAGCAGACGTGCGAGTACG mismatch AGCAGAC 64
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 65
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 hp. BC[G]=2, shift=2 GCGAGCAGACGTGCGAGTACG mismatch → AGCAGAC 66
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 67
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 hp. BC[A]=1, shift=1 GCGAGCAGACGTGCGAGTACG mismatch AGCAGAC 68
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 69
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG match → AGCAGAC 70
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG match → AGCAGAC 71
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 hp. BC[C]=4, shift=4 GCGAGCAGACGTGCGAGTACG mismatch → AGCAGAC 72
Example • Text string T=GCGAGCAGACGTGCGAGTACG • Pattern string P=AGCAGAC hp. BC A C G T 1 4 2 7 GCGAGCAGACGTGCGAGTACG → AGCAGAC 73
Time complexity • preprocessing phase in O(m+ σ) time and O(σ) space complexity, σ is the number of alphabets in pattern. • searching phase in O(mn) time complexity. 74
Quick Search Algorithm A very fast substring search algorithm, SUNDAY D. M. , Communications of the ACM. 33(8), 1990, pp. 132 -142. 75
Introduction • simplification of the Boyer-Moore algorithm. • uses only the bad-character shift. • easy to implement. • uses Rule 2 -2: 1 -Suffix Rule 76
Quick Search Rule Suppose that P 1 is aligned to Ts now, and we perform a pairwise comparing between text T and pattern P from left to right. Assume that the first mismatch occurs when comparing Tq with Pp. Since Tq ≠Pp , we move the pattern P to right such that the largest position i in the right of Pi is equal to Ts+m. We can shift the pattern at least (m-i) positions right. s q s+m T t y x mismatch P t z p 1 Shift P x i t 1 z p x i 77
Quick Search Preprocessing Table • The only thing we want to do is to construct a table as follow. Let x be a character in the alphabet. We record the position of the last x, if it exists in P, we counted the position of x from the right end. If x does not exist in P, we record it as m+1. 78
Quick Search Preprocessing Table • Example： 7654321 P=CAGAGAG qs. BC A 2 C 7 G 1 T 8 • With this table, the number of steps which we move the pattern can be easily done. After the movement, we compare the pattern and the text from left to right until a mismatch occurs, otherwise we output the position of the first character in T which aligns to pattern P. 79
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 GCGCAGAGAGTACG CAGAGAG 80
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 GCGCAGAGAGTACG mismatch CAGAGAG 81
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 qs. BC[G]=1, shift=1 GCGCAGAGAGTACG mismatch CAGAGAG 82
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 GCGCAGAGAGTACG CAGAGAG 83
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 GCGCAGAGAGTACG mismatch CAGAGAG 84
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 qs. BC[A]=2, shift=2 GCGCAGAGAGTACG mismatch CAGAGAG 85
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 GCGCAGAGAGTACG CAGAGAG 86
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 GCGCAGAGAGTACG exact match CAGAGAG 87
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 qs. BC[T]=8, shift=8 GCGCAGAGAGTACG exact match CAGAGAG 88
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 GCGCAGAGAGTACG CAGAGAG 89
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 GCGCAGAGAGTACG mismatch CAGAGAG 90
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 qs. BC[A]=2, shift=2 GCGCAGAGAGTACG mismatch CAGAGAG 91
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 GCGCAGAGAGTACG CAGAGAG 92
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 GCGCAGAGAGTACG mismatch CAGAGAG 93
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 qs. BC[G]=1, shift=1 GCGCAGAGAGTACG mismatch CAGAGAG 94
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 GCGCAGAGAGTACG CAGAGAG 95
Example • Text string T=GCGCAGAGAGTACG • Pattern string P=CAGAGAG qs. BC A C G T 2 7 1 8 GCGCAGAGAGTACG mismatch CAGAGAG 96
Time complexity • preprocessing phase in O(m+ σ) time and O(σ) space complexity, σ is the number of alphabets in pattern. • searching phase in O(mn) time complexity. 97
Smith Algorithm Experiments with a very fast substring search algorithm, SMITH P. D. , Software Practice & Experience 21(10), 1991, pp. 1065 -1074. 98
Introduction • takes the maximum of the Horspool shift function and the Quick Search shift function. • uses Rule 2 -2: 1 -Suffix Rule 99
Smith Algorithm • This algorithm is almost the same as Quick Search Algorithm except the last character of the window is also considered. If this will induce a better movement than the Quick Search Algorithm. This is used; otherwise the Quick Search is used. 100
Example • Text string T=GCGCAGAGAGTACG A C G • Pattern string hp. BC 1 6 2 P=CAGAGAG qs. BC T 7 A C G T 2 7 1 8 GCGCAGAGAGTACG CAGAGAG 101
Example • Text string T=GCGCAGAGAGTACG A C G • Pattern string hp. BC 1 6 2 P=CAGAGAG qs. BC T 7 A C G T 2 7 1 8 GCGCAGAGAGTACG mismatch CAGAGAG 102
Example • Text string T=GCGCAGAGAGTACG A C G • Pattern string hp. BC 1 6 2 P=CAGAGAG qs. BC T 7 A C G T 2 7 1 8 hp. BC[A]=1, qs. BC[G]=1, shift=1 GCGCAGAGAGTACG mismatch CAGAGAG 103
Example • Text string T=GCGCAGAGAGTACG A C G • Pattern string hp. BC 1 6 2 P=CAGAGAG qs. BC T 7 A C G T 2 7 1 8 GCGCAGAGAGTACG CAGAGAG 104
Example • Text string T=GCGCAGAGAGTACG A C G • Pattern string hp. BC 1 6 2 P=CAGAGAG qs. BC T 7 A C G T 2 7 1 8 GCGCAGAGAGTACG mismatch CAGAGAG 105
Example • Text string T=GCGCAGAGAGTACG A C G • Pattern string hp. BC 1 6 2 P=CAGAGAG qs. BC T 7 A C G T 2 7 1 8 hp. BC[G]=2, qs. BC[A]=2, shift=2 GCGCAGAGAGTACG mismatch CAGAGAG 106
Example • Text string T=GCGCAGAGAGTACG A C G • Pattern string hp. BC 1 6 2 P=CAGAGAG qs. BC T 7 A C G T 2 7 1 8 GCGCAGAGAGTACG CAGAGAG 107
Example • Text string T=GCGCAGAGAGTACG A C G • Pattern string hp. BC 1 6 2 P=CAGAGAG qs. BC T 7 A C G T 2 7 1 8 GCGCAGAGAGTACG exact match CAGAGAG 108
Example • Text string T=GCGCAGAGAGTACG A C G • Pattern string hp. BC 1 6 2 P=CAGAGAG qs. BC T 7 A C G T 2 7 1 8 hp. BC[G]=2, qs. BC[T]=8, shift=8 GCGCAGAGAGTACG exact match CAGAGAG 109
Example • Text string T=GCGCAGAGAGTACG A C G • Pattern string hp. BC 1 6 2 P=CAGAGAG qs. BC T 7 A C G T 2 7 1 8 GCGCAGAGAGTACG CAGAGAG 110
Example • Text string T=GCGCAGAGAGTACG A C G • Pattern string hp. BC 1 6 2 P=CAGAGAG qs. BC T 7 A C G T 2 7 1 8 GCGCAGAGAGTACG mismatch CAGAGAG 111
Example • Text string T=GCGCAGAGAGTACG A C G • Pattern string hp. BC 1 6 2 P=CAGAGAG qs. BC T 7 A C G T 2 7 1 8 hp. BC[T]=7, qs. BC[A]=2, shift=7 GCGCAGAGAGTACG mismatch CAGAGAG 112
Example • Text string T=GCGCAGAGAGTACG A C G • Pattern string hp. BC 1 6 2 P=CAGAGAG qs. BC T 7 A C G T 2 7 1 8 GCGCAGAGAGTACG CAGAGAG 113
Time complexity • preprocessing phase in O(m+ σ) time and O(σ) space complexity, σ is the number of alphabets in pattern. • searching phase in O(mn) time complexity. 114
Zhu-Takaoka Algorithm On improving the average case of the Boyer. Moore string matching algorithm, R. F. ZHU and T. TAKAOKA, Journal of Information Processing 10(3), 1987, pp. 173 -177 115
• The Zhu-Takaoka Algorithm is a variant of the Boyer and Moore Algorithm. The algorithm only improve the bad character of the Boyer and Moore Algorithm. • Zhu and Takaoka modified the BM Algorithm. They replaced the bad character rule by a 2 -substring rule. The good suffix rules are still used. 116
Rule 2 -3: The 2 -Substring Rule (A Special Version of Rule 2) • Consider the 2 -substring Tk and Tk+1. We may apply Rule 2 -3 now. Tk Tk+1 u x Pj u x Pi Pi+1 v x u x v x 117
Zhu-Takaoka Preprocessing Table The preprocessing phase of the algorithm consists in computing for each pair of characters (a, b) with a, b the rightmost occurrence of ab in x [ 0. . m -2] 118
![Case 1 : Þ If zt. Bc[A, C] = k • Example 0 1](http://slidetodoc.com/presentation_image_h2/3c67324f7a48362913f3b119694d967e/image-119.jpg "Case 1 : Þ If zt. Bc[A, C] = k • Example 0 1")
Case 1 : Þ If zt. Bc[A, C] = k • Example 0 1 2 3 4 5 6 7 8 9 10 11 12 Text G C A T C G C A G A G T A C G Pattern G C A G A Shift by 5 G A G G C A zt. Bc A C G * A 8 8 2 8 C 5 8 7 8 G 1 6 7 8 * 8 8 7 8 ↑ a ←b G A 13 14 15 16 4 5 6 17 18 19 20 21 22 G i 0 1 2 3 7 x[i ] G C A G A G • zt. Bc[C, A] = 5 ; k ≤ m-2 ; ∵ x[8 -5 -2. . 8 -51] = ab (x[1. . 2] = CA) and “CA” does not occur in x[8 -5 -1. . 8 -2] (x[2. . 6] ). 119 23
![Case 2 : => If zt. Bc[A, C] = k • Example 0 Text](http://slidetodoc.com/presentation_image_h2/3c67324f7a48362913f3b119694d967e/image-120.jpg "Case 2 : => If zt. Bc[A, C] = k • Example 0 Text")
Case 2 : => If zt. Bc[A, C] = k • Example 0 Text Pattern 1 2 3 4 5 6 7 8 9 10 11 12 13 14 G A G 3 15 16 17 18 19 20 21 22 G C A T C G G A G T A C G G C A G A Shift by 7 G zt. Bc A C G * A 8 8 2 8 C 5 8 7 8 G 1 6 7 8 * 8 8 7 8 ↑ a G C ←b A G A i 0 1 2 4 5 6 7 x[i ] G C A G A G • zt. Bc[C, G] = 7 ; k = m-1 ; ∵ x[0] = b ( G = G) and “CG” does not occur in x[0. . 8 -2] (x[0. . 6] ). 120 23
![Case 3 : => If zt. Bc[A, C] = k • Example 0 1](http://slidetodoc.com/presentation_image_h2/3c67324f7a48362913f3b119694d967e/image-121.jpg "Case 3 : => If zt. Bc[A, C] = k • Example 0 1")
Case 3 : => If zt. Bc[A, C] = k • Example 0 1 2 3 4 5 6 7 8 9 10 11 12 Text G C A T C G C A G A G T A C G Pattern G C A G A 13 14 15 16 18 19 20 21 22 G zt. Bc A C G * i 0 1 2 A 8 8 2 8 G C A G A G C 5 8 7 8 x[i ] G 1 6 7 8 * 8 8 7 8 ↑ a 17 ←b 3 4 5 6 7 • zt. Bc[A, C] = 8 ; k = m ; ∵ x[0] ≠b (G≠C) and “AC” does not occur in x[0. . 8 -2] ( x[0. . 6] ). 121 23
preprocessing phase Consider text= ATTGCCTAATA and pattern=CTAAG The alphabet of pattern is {A. C. G. T }; The sign “ * ” denotes a word of text which never appears in pattern. First, we fill in the blanks with the length m of pattern. Example: A C G T * A 5 5 5 C 5 5 5 G 5 5 5 T 5 5 5 * 5 5 5 122
preprocessing phase Then, we suppose the last 2 -substring ab does not occur in [0. . m-2]. If P 0 = b, we set zt. Bc[i , b] = m-1 for all i. Example: A C G T * ←b A 5 4 5 5 5 C 5 4 5 5 5 G 5 4 5 5 5 T 5 4 5 5 5 * 5 4 5 5 5 ↑ a T: ATTGCCTAAGTA P: CTAAG 123
![preprocessing phase Finally, we set zt. BC[a, b] = k if k≤ m-2 and](http://slidetodoc.com/presentation_image_h2/3c67324f7a48362913f3b119694d967e/image-124.jpg "preprocessing phase Finally, we set zt. BC[a, b] = k if k≤ m-2 and")
preprocessing phase Finally, we set zt. BC[a, b] = k if k≤ m-2 and P[m-k-2. . m-k-1]=ab and ab does not occur in P[m-k-1. . m-2]. Example: A C G T * ←b A 1 4 5 5 5 C 5 4 5 3 5 G 5 4 5 5 5 T 2 4 5 5 5 * 5 4 5 5 5 ↑ a P: CTAAG 1 2 3 124
• Full Example 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Text G C A T C G C A G A G T A C G Pattern G C A G A G A G Shift by 5 In the step, we select the zt. Bc function to shift because zt. Bc[P 6 P 7=CA] = 5 ＞ bm. Gs [7] =1. The pattern shifts 5 steps right by case 1. i x[i] bm. Gs zt. Bc A C G * 7 A 8 8 2 8 G C A G A G C 5 8 7 8 G 1 6 7 8 * 8 8 7 8 0 7 1 7 2 7 3 2 4 7 5 4 6 7 1 ↑ a ←b 125
• Full Example Text 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 G C A T C G C A G A G T A C G G C A G A G Pattern Shift by 7 exact matching G C A G In the step, we select the bm. Gs function to shift because zt. Bc[A, G] = 2 ＜ bm. Gs [0] = 7. zt. Bc A C G * i 0 1 2 3 4 5 6 7 A 8 8 2 8 x[i] G C A G A G C 5 8 7 8 G 1 6 7 8 * 8 8 7 8 bm. Gs 7 7 7 2 7 4 7 1 ↑ a ←b 126
• Full Example Text 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 G C A T C G C A G A G T A C G G C A G A G A G Pattern Shift by 4 In the step, we select the bm. Gs function to shift because zt. Bc[A, G] = 2 ＜ bm. Gs [5] = 4. i 0 1 2 3 4 5 6 7 x[i] G C A G A G bm. Gs 7 7 7 2 7 4 7 1 zt. Bc A C G * A 8 8 2 8 C 5 8 7 8 G 1 6 7 8 * 8 8 7 8 ↑ a ←b 127
• Full Example Text 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 G C A T C G C A G A G T A C G G C A G A G Pattern By the bm. Gs or zt. Bc function ; We can select the zt. Bc function or the bm. Gs function to shift because zt. Bc[C, G] = 7 = bm. Gs [6]. i 0 1 2 3 4 5 6 7 x[i] G C A G A G bm. Gs 7 7 7 2 7 4 7 1 zt. Bc A C G * A 8 8 2 8 C 5 8 7 8 G 1 6 7 8 * 8 8 7 8 ↑ a ←b 128
Time complexity • preprocessing phase in O(m＋ ) time and space complexity. Ø( = the numbers of alphabet of the text ). • searching phase in O(m × n) time complexity. 129
![Reference [KMP 77] Fast pattern matching in strings, D. E. Knuth, J. H. Morris,](http://slidetodoc.com/presentation_image_h2/3c67324f7a48362913f3b119694d967e/image-130.jpg "Reference [KMP 77] Fast pattern matching in strings, D. E. Knuth, J. H. Morris,")
Reference [KMP 77] Fast pattern matching in strings, D. E. Knuth, J. H. Morris, Jr and V. B. Pratt, SIAM J. Computing, 6, 1977, pp. 323– 350. [BM 77] A fast string search algorithm, R. S. Boyer and J. S. Moore, Comm. ACM, 20, 1977, pp. 762– 772. [S 90] A very fast substring search algorithm, D. M. Sunday, Comm. ACM, 33, 1990, pp. 132– 142. [RR 89] The Rand MH Message Handling system: User’s Manual (UCIVersion), M. T. Rose and J. L. Romine, University of California, Irvine, 1989. [S 82] A comparison of three string matching algorithms, G. De V. Smith, Software—Practice and Experience, 12, 1982, pp. 57– 66. [HS 91] Fast string searching, HUME A. and SUNDAY D. M. , Software - Practice & Experience 21(11), 1991, pp. 1221 -1248. [S 94] String Searching Algorithms , Stephen, G. A. , World Scientific, 1994. [ZT 87] On improving the average case of the Boyer-Moore string matching algorithm, ZHU, R. F. and TAKAOKA, T. , Journal of Information Processing 10(3) , 1987, pp. 173 -177. [R 92] Tuning the Boyer-Moore-Horspool string searching algorithm, RAITA T. , Software - Practice & Experience, 22(10) , 1992, pp. 879 -884. [S 94] On tuning the Boyer-Moore-Horspool string searching algorithms, SMITH, P. D. , Software - Practice & Experience, 24(4) , 1994, pp. 435 -436. [BR 92] Average running time of the Boyer-Moore-Horspool algorithm, BAEZA-YATES, R. A. , RÉGNIER, M. , Theoretical Computer Science 92(1) , 1992, pp. 19 -31. [H 80] Practical fast searching in strings, HORSPOOL R. N. , Software - Practice & Experience, 10(6) , 1980, pp. 501 -506. [L 95] Experimental results on string matching algorithms, LECROQ, T. , Software - Practice & Experience 25(7) 130 , 1995, pp. 727 -765.
Thanks for your listening 131
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