eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-07-04
62:1
62:13
10.4230/LIPIcs.ICALP.2018.62
article
Towards Unified Approximate Pattern Matching for Hamming and L_1 Distance
Gawrychowski, Pawel
1
Uznanski, Przemyslaw
2
Institute of Computer Science, University of Wrocław, Poland
Department of Computer Science, ETH Zürich, Switzerland
Computing the distance between a given pattern of length n and a text of length m is defined as calculating, for every m-substring of the text, the distance between the pattern and the substring. This naturally generalizes the standard notion of exact pattern matching to incorporate dissimilarity score. For both Hamming and L_{1} distance only relatively slow O~(n sqrt{m}) solutions are known for this generalization. This can be overcome by relaxing the question. For Hamming distance, the usual relaxation is to consider the k-bounded variant, where distances exceeding k are reported as infty, while for L_{1} distance asking for a (1 +/- epsilon)-approximation seems more natural. For k-bounded Hamming distance, Amir et al. [J. Algorithms 2004] showed an O~(n sqrt{k}) time algorithm, and Clifford et al. [SODA 2016] designed an O~((m+k^{2})* n/m) time solution. We provide a smooth time trade-off between these bounds by exhibiting an O~((m+k sqrt{m})* n/m) time algorithm. We complement the trade-off with a matching conditional lower bound, showing that a significantly faster combinatorial algorithm is not possible, unless the combinatorial matrix multiplication conjecture fails. We also exhibit a series of reductions that together allow us to achieve essentially the same complexity for k-bounded L_1 distance. Finally, for (1 +/- epsilon)-approximate L_1 distance, the running time of the best previously known algorithm of Lipsky and Porat [Algorithmica 2011] was O(epsilon^{-2} n). We improve this to O~(epsilon^{-1}n), thus essentially matching the complexity of the best known algorithm for (1 +/- epsilon)-approximate Hamming distance.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol107-icalp2018/LIPIcs.ICALP.2018.62/LIPIcs.ICALP.2018.62.pdf
approximate pattern matching
conditional lower bounds
L_1 distance
Hamming distance