Pattern matching under polynomial transformation

Ayelet Butman, Clifford Peter, Raphael Clifford, Markus T Jalsenius, Noa Lewenstein, Benny Porat, Ely Porat

Research output: Contribution to journalArticle (Academic Journal)peer-review

9 Citations (Scopus)
387 Downloads (Pure)


We consider a class of pattern matching problems where a normalising transformation is applied at every alignment. Normalised pattern matching plays a key role in fields as diverse as image processing and musical information processing where application specific transformations are often applied to the input. By considering the class of polynomial transformations of the input, we provide fast algorithms and the first lower bounds for both new and old problems. Given a pattern of length m and a longer text of length n where both are assumed to contain integer values only, we first show O(n log m) time algorithms for pattern matching under linear transformations even when wildcard symbols can occur in the input. We then show how to extend the technique to polynomial transformations of arbitrary degree. Next we consider the problem of finding the minimum Hamming distance under polynomial transformation. We show that, for any epsilon>0, there cannot exist an O(n m^(1-epsilon)) time algorithm for additive and linear transformations conditional on the hardness of the classic 3SUM problem. Finally, we consider a version of the Hamming distance problem under additive transformations with a bound k on the maximum distance that need be reported. We give a deterministic O(nk log k) time solution which we then improve by careful use of randomisation to O(n sqrt(k log k) log n) time for sufficiently small k. Our randomised solution outputs the correct answer at every position with high probability.
Original languageEnglish
Pages (from-to)611-633
Number of pages23
JournalSIAM Journal on Computing
Issue number2
Early online date3 Apr 2013
Publication statusPublished - Apr 2013


  • algorithms
  • string matching
  • lower bounds
  • 3-sum


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