Abstract
We study the problem of finding a longest common increasing subsequence (LCIS) of multiple sequences of numbers. The LCIS problem is a fundamental issue in various application areas, including the whole genome alignment and pattern recognition. In this paper we give an efficient algorithm to find the LCIS of two sequences in O(min(r logℓ, nℓ + r)loglog n + n log n) time where n is the length of each sequence and r is the total number of ordered pairs of positions at which the two sequences match and ℓ is the length of the LCIS. For m sequences where m ≥ 3, we find the LCIS in O(min(mr 2,mr log ℓlogm r) + mnlog n) time where r is the total number of m-tuples of positions at which the m sequences match. The previous results find the LCIS of two sequences in O(n 2) and O(n ℓ log n) time. Our algorithm is faster when r is relatively small, e.g., for r < min(n 2/loglogn,nℓ).
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References
Bespamyatnikh, S., Segal, M.: Enumerating longest increasing subsequences and patience sorting. Inf. Process. Lett. 76(1-2), 7–11 (2000)
Bhat, D.N.: An evolutionary measure for image matching. In: ICPR 1998: Proc. of the 14th International Conference on Pattern Recognition, vol. 1, pp. 850–852. IEEE Computer Society, Los Alamitos (1998)
Delcher, A.L., Kasif, S., Fleischmann, R.D., Peterson, J., White, O., Salzberg, S.L.: Alignment of whole genomes. Nucleic Acids Res. 27, 2369–2376 (1999)
Hunt, J.W., Szymanski, T.G.: A fast algorithm for computing longest common subsequences. Commun. ACM 20(5), 350–353 (1977)
Katriel, I., Kutz, M.: A faster algorithm for computing a longest common increasing subsequence. Research Report MPI-I-2005-1-007, Max-Planck-Institut für Informatik, Stuhlsatzenhausweg 85, 66123 Saarbrücken, Germany (March 2005)
Knuth, D.E.: Sorting and Searching, The Art of Computer Programming, vol. 3. Addison-Wesley, Reading (1973)
Kurtz, S., Phillippy, A., Delcher, A., Smoot, M., Shumway, M., Antonescu, C., Salzberg, S.: Versatile and open software for comparing large genomes. Genome Biology 5(2) (2004)
Maier, D.: The complexity of some problems on subsequences and supersequences. J. ACM 25(2), 322–336 (1978)
Marcolino, A., Ramos, V., Ramalho, M., Caldas Pinto, J.R.: Line and word matching in old documents. In: Proc. of SIARP 2000 - 5th IberoAmerican Symposium on Pattern Recognition, pp. 123–135 (2000)
Masek, W.J., Paterson, M.: A faster algorithm computing string edit distances. J. Comput. Syst. Sci. 20(1), 18–31 (1980)
Schensted, C.: Longest increasing and decreasing subsequences. Canad. J. Math. 13, 179–191 (1961)
van Emde Boas, P.: Preserving order in a forest in less than logarithmic time. In: Proc. of the 16th Symposium on Foundations of Computer Science (FOCS), pp. 75–84 (1975)
Willard, D.E., Lueker, G.S.: Adding range restriction capability to dynamic data structures. J. ACM 32(3), 597–617 (1985)
Yang, I.-H., Huang, C.-P., Chao, K.-M.: A fast algorithm for computing a longest common increasing subsequence. Inf. Process. Lett. 93(5), 249–253 (2005)
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Chan, WT., Zhang, Y., Fung, S.P.Y., Ye, D., Zhu, H. (2005). Efficient Algorithms for Finding a Longest Common Increasing Subsequence. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_67
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DOI: https://doi.org/10.1007/11602613_67
Publisher Name: Springer, Berlin, Heidelberg
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