Linear Algorithm for a Cyclic Graph Transformation
We propose a linear time and linear space algorithm that constructs a minimal (in the total cost) sequence of operations required to transform a directed graph consisting of disjoint cycles into any graph of the same type. The following operations are allowed: double-cut-and-join of vertices and insertion or deletion of a connected fragment of edges; the latter two operations have the same cost. We present a complete proof that the algorithm finds the corresponding minimum. The problem is a nontrivial particular case of the general problem of transforming a graph into another, which in turn is an instance of a hard optimization problem in graphs. In this general problem, which is known to be NP-complete, each vertex of a graph is of degree 1 or 2, edges with the same name may repeat unlimitedly, and operations belong to a well-known list including the above-mentioned operations. We describe our results for the general problem, but the proof is given for the cyclic case only.
Keywords and phrasesgraph cycle graph rearrangement operation cost combinatorial problem optimization in graphs linear algorithm
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- 3.K. Yu. Gorbunov and V. A. Lyubetsky, “A linear algorithm of graph reconfiguration,” Autom. Remote Control, No. 12 (2018, in press).Google Scholar
- 5.V. A. Lyubetsky, R. A. Gershgorin, A. V. Seliverstov, and K. Yu. Gorbunov, “Algorithms for reconstruction of chromosomal structures,” BMC Bioinform. 17, 40. 1–40. 23 (2016). doi 10.1186/s12859-016-0878-zGoogle Scholar
- 6.V. A. Lyubetsky, R. A. Gershgorin, and K. Yu. Gorbunov, “Chromosome structures: reduction of certain problems with unequal gene content and gene paralogs to integer linear programming,” BMC Bioinform. 18, 537. 1–537. 18 (2017). doi 10.1186/s12859-017-1944-xGoogle Scholar
- 8.Models and Algorithms for Genome Evolution, Ed. by C. Chauve, N. El-Mabrouk, and E. Tannier, Comput. Biol. Series (Springer, London, 2013).Google Scholar
- 10.R. A. Gershgorin, K. Yu. Gorbunov, O. A. Zverkov, L. I. Rubanov, A. V. Seliverstov, and V. A. Lyubetsky, “Highly conserved elements and chromosome structure evolution in mitochondrial genomes in ciliates,” Life 7, 9. 1–9. 11 (2017). doi 10.3390/life7010009Google Scholar
- 12.P. H. da Silva, R. Machado, S. Dantas, and M. D. V. Braga, “DCJ-indel and DCJ-substitution distances with distinct operation costs,” Algorithms Mol. Biol. 8, 21. 1–21. 15 (2013). doi 10.1186/1748-7188-8-21Google Scholar
- 13.P. E. C. Compeau, “DCJ-indel sorting revisited,” Algorithms Mol. Biol. 8, 6. 1–6. 9 (2013). doi 10.1186/1748-7188-8-6Google Scholar
- 15.S. Hannenhalli and P. A. Pevzner, “Transforming men into mice (polynomial algorithm for genomic distance problem),” in Proceedings of the 36th Annual Symposiumon Foundations of Computer Science—FOCS 1995, Milwaukee, USA, Oct. 23–25, 1995, pp. 581–592.Google Scholar
- 16.G. Li, X. Qi, X. Wang, and B. Zhu, “A linear-time algorithm for computing translocation distance between signed genomes,” in Proceedings of 15th Annual Symposium on Combinatorial Pattern Matching—CPM 2004, July 5–7, 2004, Istanbul, Turkey, Lect. Notes Comput. Sci. 3109, 323–332 (2004). doi 10.1007/978-3-540-27801-6_24zbMATHGoogle Scholar