This document discusses string matching algorithms and their complexity. It introduces the string matching problem of finding all valid shifts where a pattern occurs in a text. It describes the naive algorithm that checks for a match between the pattern and text at each possible shift in O((n-m+1)m) time. It also mentions more advanced algorithms like the Knuth-Morris-Pratt algorithm and using finite automata that have better time complexities.

1. String Matching Algorithms
Kamalesh Karmakar
Assistant Professor,
Dept. of CSE & IT,
Meghnad Saha Institute of Technology.

2. String Matching Algorithms & Complexity
Weformalizethestring-matchingproblemasfollows.WeassumethatthetextisanarrayT[1..n]oflengthnandthatthepatternisanarrayP[1..m]oflengthm≤n.WefurtherassumethattheelementsofPandTarecharactersdrawnfromafinitealphabetΣ. Forexample,wemayhaveΣ={0,1}orΣ={a,b,...,z}.ThecharacterarraysPandTareoftencalledstringsofcharacters.

3. String Matching Algorithms
WesaythatpatternPoccurswithshiftsintextT(or,equivalently,thatpatternPoccursbeginningatpositions+1intextT)if0≤s≤n−mandT[s+1..s+m]=P[1..m](thatis,ifT[s+j]=P[j],for1≤j≤m).
IfPoccurswithshiftsinT,thenwecallsavalidshift;otherwise,wecallsaninvalidshift.Thestring-matchingproblemistheproblemoffindingallvalidshiftswithwhichagivenpatternPoccursinagiventextT.Figure32.1illustratesthesedefinitions.

4. Complexity comparison of Different Algorithms

5. The Naive string-matching algorithm
ThenaivealgorithmfindsallvalidshiftsusingaloopthatcheckstheconditionP[1..m]=T[s+1..s+m]foreachofthen−m+1possiblevaluesofs.
Procedure NAIVE-STRING-MATCHER takes time O((n −m + 1)m), and this bound is tight in the worst case.

6. String matching with finite automata

7. String matching with finite automata

8. String matching with finite automata

9. String matching with finite automata

10. The Knuth-Morris-Pratt matching algorithm

11. The Knuth-Morris-Pratt matching algorithm

13. Example

14. Q & A