1. 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

2. 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

3. 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.

4. 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

5. 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

6. 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 a
A C G *
HpBc[a]

7. 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

8. 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

9. Example(1/3) GCATCGCAGAGAGTATACAGTACG GCAGAGAG
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 *

10. Example(2/3) GCATCGCAGAGAGTATACAGTACG GCAGAGAG
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 *

11. Example(3/3) GCATCGCAGAGAGTATACAGTACG GCAGAGAG A C G * 1 6 2 8
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 *

12. Example(1/2) for example : T : AGATACGATATATAC P : ATATA
HoBc[a]
AGATACGATATATAC 
ATATA 
d[A] = 2
ATATA
G ≠A, d[G] = 5

13. Example(2/2) AGATACGATATATAC ATATA
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

14. 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)

15. 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 , 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 , 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 , CRC Press Inc., Boca Raton, FL.
HANCART, C., 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):
LECROQ, T., 1995, Experimental results on string matching algorithms, Software - Practice & Experience 25(7):
STEPHEN, G.A., 1994, String Searching Algorithms, World Scientific.

16. THANK YOU