String Matching in Lempel-Ziv Compressed StringsM. Farach and M. Thorup 2 Algorithmica (1998) 20: 388-404

Preliminaries

prefix, substring, suffix ()

prefix, substring, suffix ()w = nobinobita

Pattern Matching

Pattern Matching

Data Compression 0.0000000001% 400%453 3

Data Compressionaldoghqu3850pcxps;lafdjaeqw09bjzpafq05^@62:vzZIAPF(90rwDEVcx0832nkvl;pzp99OPF:eDfja

Goal of this paper

Ideaaldoghqu3850pcxps;lafdjaeqw09bjzpafq05^@62:vzZIAPF(90rwDEVcx0832nkvl;pzp99OPF:eDfja 0.0000000001% 400%453 3

Ideaaldoghqu3850pcxps;lafdjaeqw09bjzpafq05^@62:vzZIAPF(90rwDEVcx0832nkvl;pzp99OPF:eDfja

Previous studiesEilam-Tsoreff and VishkinAmir, Landau, and VishikinAmir and BensonFarach and ThorupGasieniec, et al.Amir, Benson and FarachKarpinski, et al.Miyazaki, et al.

Kida, et al.yearresearchercompression methodrun-lengthtwo-dimensionalrun-lengthLZ77LZ77LZWstraight-line programsstraight-line programsLZW198819921992199519961996199719971998Kida

LZ77 CompressionLZ77

contents

example Z = a b c

useful propertyb a c a

Main Algorithm

Basic Idea existence problem

Basic Ideai prefix substring i suffix substring i substring Yes

Basic IdeaPattern: b a c ai i+1 a b c c b c b a c a c a c b a bb a c a

Winding Phase

Winding Phase

Winding Phase

Winding Phase

Winding Phase

Winding Phase

Winding Phase

Winding Phase

Winding Phasea b a b c a b a b c b a c a b a b b c a b i = 3

Unwinding Phase

Unwinding Phasea b ca b a b c a b a b c b a c a b a b b c a b Pattern: b a c a

Unwinding PhasePattern: b a c aa b a b c a b a b c b a c a b a b b c a b

Unwinding PhasePattern: b a c aa b a b c a b a b c b a c a b a b b c a b

Unwinding PhasePattern: b a c aa b a b c a b a b c b a c a b a b b c a b 55

Unwinding PhasePattern: b a c aa b a b c a b a b c b a c a b a b b c a b

Unwinding PhasePattern: b a c aa b a b c a b a b c b a c a b a b b c a b 6

Unwinding PhasePattern: b a c aa b a b c a b a b c b a c a b a b b c a b

Unwinding PhasePattern: b a c aa b a b c a b a b c b a c a b a b b c a b

Unwinding PhasePattern: b a c aa b a b c a b a b c b a c a b a b b c a b

Final operation

Complexity of Algorithm

Winding Phase

Winding PhaseL :O( log | Z | )Balanced tree

Winding PhaseL :O( log | Z | )O( 1 )

Winding PhaseL :O( | Z | log |T | )Segment-Merge

Winding Phase

Unwinding PhasePattern: b a c aa b a b c a b a b c b a c a b a b b c a b 55

Unwinding PhasePattern: b a c aa b a b c a b a b c b a c a b a b b c a b O( | Z | )O( log| P | )55

Unwinding PhaseO( P + | Z | ( | Z | + log |P | ) )

final operationO( log| P | )O( log| P | )O( log| P | )O( log| P | )O( log| P | )O( log| P | )O( log| P | )O( |Z| log| P | )

Total

Conclusion

conclusion

conclusion

String Matching in Lempel-Ziv Compressed String W W X,Y,Z X W Y W Z W W W W nob,nobin W existence problem YES,NO YES all-occurrences problem KMPBM

KMPFarachThorupLZ77LZ77LZ77 0 a -3,1 b -2,1 ab 0 2 0,2 c -1,1 ababc 0 5 0,5 substring substring baca LZ77LZ77P+P- i P-i P+i+1 substring i i+1 i suffix substring P- prefix substring P+ P-i P+i+1 YES P- P+ P- P+ Winding phaseWinding PhaseWinding PhaseWinding PhaseP- cP+ a c a i=3 YES Unwinding Phase Unwinding Phase Winding Phase P+P-Unwinding PhaseUnwinding Phase

ba substring ba ba Unwinding ca substring P+ P- i 5 YESLZ77Winding Phase Balanced treeO(log|Z|)Segment-MergeO(|Z|log|T|)Winding Phase Unwinding Phase Unwinding Phase Winding PhaseWinding Phase substring substring O(|P|)O(log|P| )Unwinding PhaseP+P- substring substring T/Zlog |P|O(|P|+|Z|)(competitive and opportunistic)