1 A simple fast hybrid pattern- matching algorithm Department of Computer Science and Information Engineering National Cheng Kung University, Taiwan R.O.C. Authors: Frantisek Franek, Christopher G. Jennings, W. F. Smyth Publisher: Journal of Discrete Algorithms 2007 Present: Chung-Chan Wu Date: December 11, 2007

2 Outline Introduction Algorithm Description KMP (Knuth-Morris-Pratt) Boyer-Moore Sunday shift The Hybrid Algorithm (FJS) Extension Experimental Results Conclusions

3 Introduction This contribution resides in these categories: In an effort to reduce processing time, we propose a mixture of Sunday’s variant of BM with KMP. Our goal is to combine the best/average case advantages of Sunday’s algorithm (BMS) with the worst case guarantees of KMP According to the experiments we have conducted, our new algorithm (FJS) is among the fastest in practice for the computation of all occurrences of a pattern p = p[1..m] in a text string x = x[1..n] on an alphabet Σ of size k.

4 KMP (Knuth-Morris-Pratt) Main Feature Perform the comparisons from left to right Space and time complexity ： O(m) Searching phase ： O(m+n) A pre-compute table called pi-table to compare backward. The π value will avoid another immediate mismatch the character of the prefix in the pattern must be different from the character comparing presently. The best worst case running time in software algorithm.

5 KMP (Knuth-Morris-Pratt) index pattern[ i ]GCAGAGAG π-value Input string: pattern: GCATCGCAGAGAGTATACAGTACG GCAGAGAG

6 Boyer-Moore Main Feature Performs the comparisons from right to left Preprocessing phase ： O(m+δ) in Space and time complexity Searching phase ： O(mn) A pre-compute table called delta_1 and delta_2. Perform well in best / average case.

7 BM - Observation 1 If char is known not occur in pattern, then we know we need not consider the possibility of an occurrence of the pattern. Input string: pattern: ub ua contains no b bad-character shift b does not occur in the pattern, use δ 1 m k

8 BM - Observation 2 If the rightmost occurrence of char in pattern is δ1 characters from the right end of pattern, then we know we can slide pattern down δ1 positions without checking for matches. Input string: pattern: ub ua bad-character shift b occurs in the pattern, use δ 1 contains no bb m k

9 BM - Observation 3(a) The good-suffix shift consists in aligning the segment y[i+j+1 … j+m-1] = x[i+1 … m-1] with its rightmost occurrence in x that is preceded by a character different from x[i]. Input string: pattern: ub ua good-suffix shift u reoccurs in pattern preceded by c ≠ a, use δ 2 ~au m k

10 BM - Observation 3(b) If there exists no such a segment, the shift consists in aligning the longest suffix v of y[i+j+1 … j+m-1] with a matching prefix of x. Input string: pattern: ub ua good-suffix shift Only a suffix v of u reoccurs in pattern, use δ 2 v v m k

11 Boyer-Moore Example δ1δ1AELMPXrest shift δ2δ2EXAMPLE HEREISASIMPLEEXAMPLE EXAMPLE Input string: pattern:

12 Sunday Shift δ1δ1AELMPXrest shift Input string: pattern: prefixb mpleaxe p δ1δ1AELMPXrest shift Input string: pattern: prefixb mpleaxe p s mpleaxe Boyer Moore Sunday Shift

13 FJS Algorithm Definitions Search p = p[1..m] in x = x[1..n] by shifting p from left to right along x. position j = 1 of p is aligned with a position i ∈ 1..n − m + 1 in x partial match: if a mismatch occurs at j >1, we say that a partial match has been determined with p[1..j − 1]. i’ = i + m - j Input string: pattern: pat ii’ j m

14 FJS Algorithm Strategy Whenever no partial match of p with x[i..i + m − 1] has been found, Sunday shifts are performed to determine the next position i’ at which x[ i’ ] = p[m]. When such an i has been found, KMP matching is then performed on p[1..m− 1] and x[i −m+ 1..i − 1]. If a partial match of p with x has been found, KMP matching is continued on p[1..m]. once a suitable i’ has been found, the first half of FJS just performs KMP matching in a different order: position m of p is compared first, followed by 1, 2,..., m − 1

15 FJS Algorithm Pre-processing Sunday’s array Δ = Δ[1..k], computable in O(m + k) time. KMP array β’ = β’[1..m+1], computable in O(m) time.

16 FJS Algorithm index pattern[ i ]GCAGAGAG π-value δ1δ1ACGrest shift2719 GCATCGCAGAGAGTATACAGTACG GCAGAGAG Input string: pattern:

17 Extension The alphabet-based preprocessing arrays of BM-type algorithms are their most useful feature, but they can be a source of trouble as well. The ASCII alphabet: Text were usually of 8 bits or less. The processing time can be regardless. The natural language text: Wide characters DNA data: {A, T, C, G} is mapping into {00, 01, 10, 11}, the alphabets of size varying by powers of 2 from 2 to 64. An example DNA : ACTG The preprocessing time is a bottleneck.

18 Environment

19 Experiment – Frequency These patterns occur 3,366,899 times

20 Experiment – Pattern Length

21 Experiment – Alphabet Size

22 Experiment – Pathological Cases

23 Conclusion We have tested FJS against four high-profile competitors (BMH, BMS, RC, TBM) over a range of contexts: pattern frequency (C1), pattern length (C2), alphabet size (C3), and pathological cases (C4). FJS was uniformly superior to its competitors, with an up to 10% advantage over BMS and RC For FJS the pathological cases (C4) are those in which the KMP part is forced to execute on prefixes of patterns where KMP provides no advantage we presented a hybrid exact pattern-matching algorithm, FJS, which combines the benefits of KMP and BMS. It requires O(m + k) time and space for preprocessing