Abstract
A new distance between strings, termed rank distance, was introduced by Dinu (Fundamenta Informaticae, 2003). Since then, the properties of rank distance were studied in several papers. In this article, we continue the study of rank distance. More precisely we tackle three problems that concern the distance between strings.
-
1.
The first problem that we study is String with Fixed Rank Distance (SFRD): given a set of strings S and an integer d decide if there exists a string that is at distance d from every string in S. For this problem we provide a polynomial time exact algorithm.
-
2.
The second problem that we study is named is the Closest String Problem under Rank Distance (CSRD). The input consists of a set of strings S, asks to find the minimum integer d and a string that is at distance at most d from all strings in S. Since this problem is NP-hard (Dinu and Popa, CPM 2012) it is likely that no polynomial time algorithm exists. Thus, we propose three different approaches: a heuristic approach and two integer linear programming formulations, one of them using geometric interpretation of the problem.
-
3.
Finally, we approach the Farthest String Problem via Rank Distance (FSRD) that asks to find two strings with the same frequency of characters (i.e. the same Parikh vector) that have the largest possible rank distance. We provide a polynomial time exact algorithm for this problem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Rank distance can be defined for strings that do not necessarily have the same Parikh vector (see, e.g., [12]). However, these strings can be transformed into strings with the same Parikh vector without affecting the rank distance. Thus, for the sake of simplicity, we do not consider such strings in our paper.
References
Arbib, C., Felici, G., Servilio, M., Ventura, P.: Optimum solution of the closest string problem via rank distance. In: Cerulli, R., Fujishige, S., Mahjoub, A.R. (eds.) ISCO 2016. LNCS, vol. 9849, pp. 297–307. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-45587-7_26
Babaie, M., Mousavi, S.R.: A memetic algorithm for closest string problem and farthest string problem. In: 2010 18th Iranian Conference on Electrical Engineering. IEEE, May 2010
Bādoiu, M., Har-Peled, S., Indyk, P.: Approximate clustering via core-sets. In: Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, STOC 2002, pp. 250–257. ACM, New York (2002)
Ben-Dor, A., Lancia, G., Ravi, R., Perone, J.: Banishing bias from consensus sequences. In: Apostolico, A., Hein, J. (eds.) CPM 1997. LNCS, vol. 1264, pp. 247–261. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-63220-4_63
de la Higuera, C., Casacuberta, F.: Topology of strings: median string is NP-complete. Theor. Comput. Sci. 230(1–2), 39–48 (2000)
Deng, X., Li, G., Li, Z., Ma, B., Wang, L.: Genetic design of drugs without side-effects. SIAM J. Comput. 32(4), 1073–1090 (2003)
Deza, E., Deza, M.: Dictionary of Distances. North-Holland, Amsterdam (2006)
Dinu, A., Dinu, L.P.: On the syllabic similarities of romance languages. In: Gelbukh, A. (ed.) CICLing 2005. LNCS, vol. 3406, pp. 785–788. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-30586-6_88
Dinu, L.P.: On the classification and aggregation of hierarchies with different constitutive elements. Fundam. Inform. 55(1), 39–50 (2003)
Dinu, L.P., Ionescu, R., Tomescu, A.: A rank-based sequence aligner with applications in phylogenetic analysis. PLoS ONE 9(8), e104006 (2014)
Dinu, L.P., Manea, F.: An efficient approach for the rank aggregation problem. Theor. Comput. Sci. 359(1–3), 455–461 (2006)
Dinu, L.P., Popa, A.: On the closest string via rank distance. In: Kärkkäinen, J., Stoye, J. (eds.) CPM 2012. LNCS, vol. 7354, pp. 413–426. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31265-6_33
Dinu, L.P., Sgarro, A.: A low-complexity distance for DNA strings. Fundam. Inform. 73(3), 361–372 (2006)
Frances, M., Litman, A.: On covering problems of codes. Theory Comput. Syst. 30(2), 113–119 (1997)
Gagolewski, M.: Data Fusion: Theory, Methods, and Applications. Institute of Computer Science, Polish Academy of Sciences, Warsaw, Poland (2015)
Gramm, J., Huffner, F., Niedermeier, R.: Closest strings, primer design, and motif search. In: Currents in Computational Molecular Biology. RECOMB, pp. 74–75 (2002)
Greenhill, S.J.: Levenshtein distances fail to identify language relationships accurately. Comput. Linguist. 37(4), 689–698 (2011)
Ionescu, R.T., Popescu, M.: Knowledge Transfer between Computer Vision and Text Mining - Similarity-Based Learning Approaches. Advances in Computer Vision and Pattern Recognition. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-30367-3
Ionescu, R.T., Popescu, M., Cahill, A.: String kernels for native language identification: insights from behind the curtains. Comput. Linguist. 42(3), 491–525 (2016)
Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12(3), 415–440 (1987)
Koonin, E.V.: The emerging paradigm and open problems in comparative genomics. Bioinformatics 15(4), 265–266 (1999)
Lanctot, J.K., Li, M., Ma, B., Wang, S., Zhang, L.: Distinguishing string selection problems. Inf. Comput. 185(1), 41–55 (2003)
Lenstra, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)
Li, M., Ma, B., Wang, L.: Finding similar regions in many sequences. J. Comput. Syst. Sci. 65(1), 73–96 (2002)
Liu, X., He, H., Sýkora, O.: Parallel genetic algorithm and parallel simulated annealing algorithm for the closest string problem. In: Li, X., Wang, S., Dong, Z.Y. (eds.) ADMA 2005. LNCS (LNAI), vol. 3584, pp. 591–597. Springer, Heidelberg (2005). https://doi.org/10.1007/11527503_70
Meneses, C.N., Lu, Z., Oliveira, C.A.S., Pardalos, P.M.: Optimal solutions for the closest-string problem via integer programming. INFORMS J. Comput. 16(4), 419–429 (2004)
Nerbonne, J., Hinrichs, E.W.: Linguistic distances. In: Proceedings of the Workshop on Linguistic Distances, Sydney, July 2006, pp. 1–6 (2006)
Nicolas, F., Rivals, E.: Complexities of the centre and median string problems. In: Baeza-Yates, R., Chávez, E., Crochemore, M. (eds.) CPM 2003. LNCS, vol. 2676, pp. 315–327. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-44888-8_23
Nicolas, F., Rivals, E.: Hardness results for the center and median string problems under the weighted and unweighted edit distances. J. Discrete Algorithms 3(2–4), 390–415 (2005)
Popescu, M., Dinu, L.P.: Rank distance as a stylistic similarity. In: 22nd International Conference on Computational Linguistics, Posters Proceedings, COLING 2008, 18–22 August 2008, Manchester, UK, pp. 91–94 (2008)
Popov, V.Y.: Multiple genome rearrangement by swaps and by element duplications. Theor. Comput. Sci. 385(1–3), 115–126 (2007)
Ritter, J.: An efficient bounding sphere. In: Graphics Gems, pp. 301–303. Elsevier (1990)
Sun, Y., et al.: Combining genomic and network characteristics for extended capability in predicting synergistic drugs for cancer. Nat. Commun. 6, 8481 (2015)
Wang, L., Dong, L.: Randomized algorithms for motif detection. J. Bioinf. Comput. Biol. 3(5), 1039–1052 (2005)
Wooley, J.C.: Trends in computational biology: a summary based on a RECOMB plenary lecture. J. Comput. Biol. 6(3/4), 459–474 (1999)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Dinu, L.P., Dumitru, B.C., Popa, A. (2019). Algorithms for Closest and Farthest String Problems via Rank Distance. In: Gopal, T., Watada, J. (eds) Theory and Applications of Models of Computation. TAMC 2019. Lecture Notes in Computer Science(), vol 11436. Springer, Cham. https://doi.org/10.1007/978-3-030-14812-6_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-14812-6_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-14811-9
Online ISBN: 978-3-030-14812-6
eBook Packages: Computer ScienceComputer Science (R0)