Abstract
For two strings a, b, the longest common subsequence (LCS) problem consists in comparing a and b by computing the length of their LCS . In a previous paper, we defined a generalisation, called “the all semi-local LCS problem”, for which we proposed an efficient output representation and an efficient algorithm. In this paper, we consider a restriction of this problem to strings that are permutations of a given set. The resulting problem is equivalent to the all local longest increasing subsequences (LIS) problem. We propose an algorithm for this problem, running in time O(n 1.5) on an input of size n. As an interesting application of our method, we propose a new algorithm for finding a maximum clique in a circle graph on n nodes, running in the same asymptotic time O(n 1.5). Compared to a number of previous algorithms for this problem, our approach presents a substantial improvement in worst-case running time.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Albert, M.H., Golynski, A., Hamel, A.M., López-Ortiz, A., Rao, S.S., Safari, M.A.: Longest increasing subsequences in sliding windows. Theoretical Computer Science 321, 405–414 (2004)
Alves, C.E.R., Cáceres, E.N., Song, S.W.: An all-substrings common subsequence algorithm. Electronic Notes in Discrete Mathematics 19, 133–139 (2005)
Apostolico, A., Atallah, M.J., Hambrusch, S.E.: New clique and independent set algorithms for circle graphs. Discrete Applied Mathematics 36, 1–24 (1992)
Bentley, J.L.: Multidimensional divide-and-conquer. Communications of the ACM 23(4), 214–229 (1980)
Bespamyatnikh, S., Segal, M.: Enumerating longest increasing subsequences and patience sorting. Information Processing Letters 76, 7–11 (2000)
Chen, E., Yuan, H., Yang, L.: Longest increasing subsequences in windows based on canonical antichain partition. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 1153–1162. Springer, Heidelberg (2005)
Dijkstra, E.W.: Some beautiful arguments using mathematical induction. Acta Informatica 13, 1–8 (1980)
Even, S., Itai, A.: Queues, stacks and graphs. In: Theory of Machines and Computations, pp. 71–86. Academic Press, London (1971)
Gavril, F.: Algorithms for a maximum clique and a maximum independent set of a circle graph. Networks 3, 261–273 (1973)
Hsu, W.-L.: Maximum weight clique algorithms for circular-arc graphs and circle graphs. SIAM Journal on Computing 14(1), 224–231 (1985)
JaJa, J., Mortensen, C., Shi, Q.: Space-efficient and fast algorithms for multidimensional dominance reporting and counting. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 558–568. Springer, Heidelberg (2004)
Masuda, S., Nakajima, K., Kashiwabara, T., Fujisawa, T.: Efficient algorithms for finding maximum cliques of an overlap graph. Networks 20, 157–171 (1990)
Rotem, D., Urrutia, J.: Finding maximum cliques in circle graphs. Networks 11, 269–278 (1981)
Tiskin, A.: All semi-local longest common subsequences in subquadratic time. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds.) CSR 2006. LNCS, vol. 3967, pp. 352–363. Springer, Heidelberg (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tiskin, A. (2006). Longest Common Subsequences in Permutations and Maximum Cliques in Circle Graphs. In: Lewenstein, M., Valiente, G. (eds) Combinatorial Pattern Matching. CPM 2006. Lecture Notes in Computer Science, vol 4009. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11780441_25
Download citation
DOI: https://doi.org/10.1007/11780441_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35455-0
Online ISBN: 978-3-540-35461-1
eBook Packages: Computer ScienceComputer Science (R0)