On Almost Monge All Scores Matrices
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The all scores matrix of a grid graph is a matrix containing the optimal scores of paths from every vertex on the first row of the graph to every vertex on its last row. This matrix is commonly used to solve diverse string comparison problems. All scores matrices have the Monge property, and this was exploited by previous works that used all scores matrices for solving various problems. In this paper, we study an extension of grid graphs that contain an additional set of edges, called bridges. Our main result is to show several properties of the all scores matrices of such graphs. We also apply these properties to obtain an \(O(r(nm+n^2))\) time algorithm for constructing the all scores matrix of an \(m\times n\) grid graph with r bridges and bounded integer weights.
KeywordsSequence alignment Longest common subsequences DIST matrices Monge matrices All path score computations Multiple-source shortest-paths
We thank the anonymous reviewers, that helped us improve the readability of the paper through their many helpful suggestions. The research of A.C and D.T was partially supported by ISF Grant No. 981/11. The research of A.C and M.Z-U was partially supported by ISF Grant 179/14.
- 4.Apostolico, A.: String editing and longest common subsequences. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 361–398. Springer (1997)Google Scholar
- 5.Arslan, A.N.: Sequence alignment guided by common motifs described by context free grammars. In: Biotechnology and Bioinformatics Symposium (BIOT-2007). Colorado Springs, CO (2007)Google Scholar
- 7.Bodlaender, H.L., Fellows, M.R., Evans, P.A.: Finite-state computability of annotations of strings and trees. In: Proceedings of the 7th Symposium on Combinatorial Pattern Matching (CPM), pp. 384–391. Springer (1996)Google Scholar
- 12.Eisenstat, D., Klein, P.N.: Linear-time algorithms for max flow and multiple-source shortest paths in unit-weight planar graphs. In: Proceedings of the 45th ACM Symposium on Theory of Computing (STOC), pp. 735–744 (2013)Google Scholar
- 13.Evans, P.A.: Algorithms and Complexity for Annotated Sequence Analysis. Ph.D. thesis, Citeseer (1999)Google Scholar
- 14.Hermelin, D., Landau, G.M., Landau, S., Weimann, O.: A unified algorithm for accelerating edit-distance computation via text-compression. In: Proceedings of the 26th Symposium on Theoretical Aspects of Computer Science (STACS), pp. 529–540 (2009)Google Scholar
- 17.Ishida, Y., Inenaga, S., Shinohara, A., Takeda, M.: Fully incremental LCS computation. In: Proceedings of the 15th Symposium on Fundamentals of Computation Theory (FCT), pp. 563–574 (2005)Google Scholar
- 21.Klein, P.N.: Multiple-source shortest paths in planar graphs. In: Proceedings of the 16th Symposium on Discrete Algorithms (SODA), vol. 5, pp. 146–155 (2005)Google Scholar
- 22.Krusche, P., Tiskin, A.: String comparison by transposition networks, In: Proceedings of the of London Algorithmics, Workshop, pp. 184–204 (2008)Google Scholar
- 23.Krusche, P., Tiskin, A.: New algorithms for efficient parallel string comparison. In: Proceedings of the 22nd Symposium on Parallel Algorithms and Architectures (SPAA), pp. 209–216 (2010)Google Scholar