Algorithms on strings, trees, and sequences: computer science and computational biology
Algorithms on strings, trees, and sequences: computer science and computational biology
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
A Survey of Longest Common Subsequence Algorithms
SPIRE '00 Proceedings of the Seventh International Symposium on String Processing Information Retrieval (SPIRE'00)
Property matching and weighted matching
Theoretical Computer Science
Journal of Discrete Algorithms
Varieties of regularities in weighted sequences
AAIM'10 Proceedings of the 6th international conference on Algorithmic aspects in information and management
String matching with swaps in a weighted sequence
CIS'04 Proceedings of the First international conference on Computational and Information Science
Approximate matching in weighted sequences
CPM'06 Proceedings of the 17th Annual conference on Combinatorial Pattern Matching
An algorithmic framework for motif discovery problems in weighted sequences
CIAC'10 Proceedings of the 7th international conference on Algorithms and Complexity
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We deal with a variant of the well-known Longest Common Subsequence (LCS) problem for weighted sequences. A (biological) weighted sequence determines the probability for each symbol to occur at a given position of the sequence (such sequences are also called Position Weighted Matrices, PWM). Two possible such versions of the problem were proposed by (Amir et al., 2009 and 2010), they are called LCWS and LCWS2 (Longest Common Weighted Subsequence 1 and 2 Problem). We solve an open problem, stated in conclusions of the paper by Amir et al., of the tractability of a log-probability version of LCWS2 problem for bounded alphabets, showing that it is NP-hard already for an alphabet of size 2. We also improve the (1/|Σ|)-approximation algorithm given by Amir et al. (where Σ is the alphabet): we show a polynomial-time approximation scheme (PTAS) for the LCWS2 problem using O(n5) space. We also give a simpler (1/2)-approximation algorithm for the same problem using only O(n2) space.