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Horspool Algorithm

Source : Practical fast searching in stringsR. NIGEL HORSPOOL

Advisor: Prof. R. C. T. LeeSpeaker: H. M. Chen

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Definition of String Matching Problem

Given a pattern string P of length m and a text string Tof length n, we would like to know whether thereexists an occurrence of P in T.

Pattern

Text

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Rule 2: Character Matching Rule

For any character X in T, find the nearest X in P which is tothe left of X in T.

T

P

x

x

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For each position of the window, we compare its lastcharacter( ) 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 textcharacter.

Text

Pattern

Suffix search

match

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Text

Safe shift

no in this part

Then, no matter whether there is a match or not, we shift thewindow so that the pattern matches . Note that is the lastcharacter of the previous window.

Text

Pattern

Suffix search

match

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Preprocessing phase

HpBc tableThe value bmBc for a particular alphabet is defined as therightmost position of that character in the pattern 1.

a A C G *

HpBc[a] 1 6 2 8

Example :T : GCATCGCAGAGAGTATACAGTACG

P : GCAGAGAG7 6 5 4 3 2 1

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Horspool (P = p1 p2p m,T =t 1t 2t n)PreprocessingForc Do d [c] m For j 1 m-1 Dod [ p j] m - j

Searching pos0 While pos n-m Do

j mWhile j > 0 And t pos+j = p j Do j j-1If j = 0 Then report an occurrence at pos+1 pos pos +d [t pos+m]

End of while

Pseudo code

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Step1:For c Do d [c ] m

c {A C G T}d[A]=8 , d[C]=8d[G]=8 , d[T]=8

Preprocessing phase

for example :T : GCATCGCAGAGAGTATACAGTACGP : GCAGAGAG

Step2:

For j 1 m- 1 Do d [p j ] m j

d[A]=1 , d[C]=6d[G]=2 , d[T]=8

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

GCATCGCAGAGAGTATACAGTACGGCAGAGAG pos 0 + d[t 0+7 ] , pos 0 + d[ A ], pos 1

GCATCGC AG AGAGTATACAGTACGGCAGAG AG

pos 1 + d[t 1+7 ] , pos 1 + d[ G ], pos 3

GCATCGC AGAG AGTATACAGTACGGCAG AGAG

pos 3 + d[t 3+7 ] , pos 3 + d[ G ], pos 5

A C G *1 6 2 8

pos pos +d[t pos +m ]

Example(1/3)

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

GCATCGCAGAGAG TATACAGTACGGCAGAGAG

While j > 0 And t pos+j = p j Do j j-1

If j = 0 Then report an occurrence at pos+1 pos 5 + d[t 5+7 ] , pos 5 + d[ G ], pos 7

GCATCGCAGAGAGTATACAGTACGGCAGAGAG

pos 7 + d[t 7+7 ] , pos 7 + d[ A ], pos 8

GCATCGCAGAGAGTATACAGTACGGCAGAGAG

pos 8 + d[t 8+7 ] , pos 8 + d[ T ], pos 16

A C G *1 6 2 8

Example(2/3)

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Example(3/3)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

GCATCGCAGAGAGTATACAGTACGGCAGAGAG

pos 16 + d[t 16+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

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

AGATACGATATA TACATATA

We verify backward the window and find the occurrence. Wethen shift by re-using the last character of the window, d[A] = 2

AGATACGATATATA CATATA

We find the pattern. We shift by the last character of thenwindow, d[A] = 2. Then, pos > n-m and the search stops.

A T *2 1 5

Example(2/2)

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Time complexity

preprocessing phase inO (m+ ) time andO () spacecomplexity.

searching phase inO (mn) time complexity.

the average number of comparisons for one text character is between 1/ and 2/(+1). ( is the number of storing characters)

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References

AHO, A.V., 1990, Algorithms for finding patterns in strings. in Handbook ofTheoretical Computer Science, Volume A, Algorithms and complexity, J. vanLeeuwen ed., Chapter 5, pp 255-300, Elsevier, Amsterdam.

BAEZA-YATES, R.A., RGNIER, M., 1992, Average running time of the Boyer-Moore-Horspool algorithm,Theoretical Computer Science 92(1):19-31.

BEAUQUIER, D., BERSTEL, J., CHRTIENNE, P., 1992, lmentsd'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, pp11-1--11-28, CRC Press Inc., Boca Raton, FL.

HANCART, C., 1993. Analyse exacte et en moyenne d'algorithmes de recherched'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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THANK YOU