Pattern Matching With Don’t Cares Clifford & Clifford’s Algorithm Orgad Keller
Orgad Keller - Algorithms 2 - Further Reading 2 Input: A text, a pattern over alphabet ( is the don’t care symbol). Output: All locations where PM with Don’t Cares
Orgad Keller - Algorithms 2 - Further Reading 3 Known Solutions We’ve seen an algorithm for the binary alphabet case. For the general alphabet case, we had two options, both based on reductions to the binary case: If we wanted only to find the matches, we had the Fischer-Paterson algorithm. If we wanted mismatch count also, we had the Abrahamson-Kosaraju algorithm.
Orgad Keller - Algorithms 2 - Further Reading 4 New Solution Sometimes simpler is better! Raphaël and Peter Clifford came up with a much, much simpler PM with Don’t Cares algorithm in time.
Orgad Keller - Algorithms 2 - Further Reading 5 Let be a symbol from the pattern and be a symbol from the text. As mentioned, In our matching relation, iff: Idea
Orgad Keller - Algorithms 2 - Further Reading 6 Since we encode as 0, it’s like saying: Idea
Orgad Keller - Algorithms 2 - Further Reading 7 As mentioned, we seek All locations where As we’ve seen, that’s true iff: Idea
Orgad Keller - Algorithms 2 - Further Reading 8 We have a problem because, generally speaking: but the opposite is not true! On the other hand, if we know that then Problem
Orgad Keller - Algorithms 2 - Further Reading 9 So we can easily fix the problem: Now: Problem
Orgad Keller - Algorithms 2 - Further Reading 10 To sum up, we got that: Problem left: How can we compute for all efficiently? Idea (cont’d)
Orgad Keller - Algorithms 2 - Further Reading 11 Computing Efficiently
Orgad Keller - Algorithms 2 - Further Reading 12 We can easily compute, and for all ‘s using FFT-based multiplication in time. Then we’ll return all locations where Convolutions
Orgad Keller - Algorithms 2 - Further Reading 13 Algorithm
Orgad Keller - Algorithms 2 - Further Reading 14 Most costly operations are 3 polynomial multiplications, so overall time is Running Time