Source : Practical fast searching in strings Horspool Algorithm Source : Practical fast searching in strings R. NIGEL HORSPOOL Advisor: Prof. R. C. T. Lee Speaker: H. M. Chen

Definition of String Matching Problem Given a pattern string P of length m and a text string T of length n, we would like to know whether there exists an occurrence of P in T. Pattern Text

Rule 2: Character Matching Rule For any character X in T, find the nearest X in P which is to the left of X in T.

For each position of the window, we compare its last character(ß) with the last character of the pattern. If they match, we scan the window backwardly against the pattern until we either find the pattern or fail on a text character. ß Text Pattern α σ Suffix search match

character of the previous window. Then, no matter whether there is a match or not, we shift the window so that the pattern matches ß. Note that ß is the last character of the previous window. ß Text Pattern α σ Suffix search match ß Text Safe shift no ß in this part

Preprocessing phase HpBc table The value bmBc for a particular alphabet is defined as the rightmost position of that character in the pattern – 1. Example : T : GCATCGCAGAGAGTATACAGTACG P : GCAGAGAG 7 6 5 4 3 2 1 a A C G * HpBc[a] 1 6 2 8

Pseudo code Horspool (P = p1p2…pm,T = t1t2…tn) Preprocessing For c  ∑ Do d[c] ← m For j  1…m-1 Do d[pj] ← m - j Searching pos←0 While pos ≤ n-m Do j ←m While j > 0 And tpos+j = pj Do j ← j-1 If j = 0 Then report an occurrence at pos+1 pos ← pos +d[tpos+m] End of while

Preprocessing phase for example : T : GCATCGCAGAGAGTATACAGTACG P : GCAGAGAG Step1: For c  ∑ Do d[c] ← m c  {A C G T} d[A]=8 , d[C]=8 d[G]=8 , d[T]=8 Step2: For j  1…m-1 Do d[pj] ← m – j d[A]=1 , d[C]=6 d[G]=2 , d[T]=8

Example(1/3) GCATCGCAGAGAGTATACAGTACG GCAGAGAG 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 GCATCGCAGAGAGTATACAGTACG  GCAGAGAG  pos ← 0 + d[t0+7] , pos ← 0 + d[A], pos ← 1 pos ← 1 + d[t1+7] , pos ← 1 + d[G], pos ← 3 GCATCGCAGAGAGTATACAGTACG pos ← 3 + d[t3+7] , pos ← 3 + d[G], pos ← 5 pos ← pos +d[tpos+m] A C G * 1 6 2 8

Example(2/3) GCATCGCAGAGAGTATACAGTACG GCAGAGAG 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 GCATCGCAGAGAGTATACAGTACG  GCAGAGAG  While j > 0 And tpos+j = pj Do j ← j-1 If j = 0 Then report an occurrence at pos+1 pos ← 5 + d[t5+7] , pos ← 5 + d[G], pos ← 7 pos ← 7 + d[t7+7] , pos ← 7 + d[A], pos ← 8 pos ← 8 + d[t8+7] , pos ← 8 + d[T], pos ← 16 A C G * 1 6 2 8

Example(3/3) GCATCGCAGAGAGTATACAGTACG GCAGAGAG A C G * 1 6 2 8 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 GCATCGCAGAGAGTATACAGTACG GCAGAGAG  pos ← 16 + d[t16+7] , pos ← 16 + d[G], pos ← 18 pos > n-m // pos >23-7 jump out of while loop A C G * 1 6 2 8

Example(1/2) for example : T : AGATACGATATATAC P : ATATA HoBc[a] 2 1 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 AGATACGATATATAC  ATATA  d[A] = 2 ATATA G ≠A, d[G] = 5

Example(2/2) AGATACGATATATAC ATATA 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 AGATACGATATATAC  ATATA We verify backward the window and find the occurrence. We then shift by re-using the last character of the window, d[A] = 2 AGATACGATATATAC ATATA   We find the pattern. We shift by the last character of then window, d[A] = 2. Then, pos > n-m and the search stops. A T * 2 1 5

Time complexity preprocessing phase in O(m+ п) time and O(п) space complexity. searching phase in O(mn) time complexity. the average number of comparisons for one text character is between 1/п and 2/(п+1). (п is the number of storing characters)

References AHO, A.V., 1990, Algorithms for finding patterns in strings. in Handbook of Theoretical Computer Science, Volume A, Algorithms and complexity, J. van Leeuwen ed., Chapter 5, pp 255-300, Elsevier, Amsterdam. BAEZA-YATES, R.A., RÉGNIER, M., 1992, Average running time of the Boyer-Moore-Horspool algorithm, Theoretical Computer Science 92(1):19-31. BEAUQUIER, D., BERSTEL, J., CHRÉTIENNE, P., 1992, Éléments d'algorithmique, Chapter 10, pp 337-377, Masson, Paris. CROCHEMORE, M., HANCART, C., 1999, Pattern Matching in Strings, in Algorithms and Theory of Computation Handbook, M.J. Atallah ed., Chapter 11, pp 11-1--11-28, CRC Press Inc., Boca Raton, FL. HANCART, C., 1993. Analyse exacte et en moyenne d'algorithmes de recherche d'un motif dans un texte, Ph. D. Thesis, University Paris 7, France. HORSPOOL R.N., 1980, Practical fast searching in strings, Software - Practice & Experience, 10(6):501-506. LECROQ, T., 1995, Experimental results on string matching algorithms, Software - Practice & Experience 25(7):727-765. STEPHEN, G.A., 1994, String Searching Algorithms, World Scientific.

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