Abstract
In this paper we address the problem of designing O(n) space representations for permutation and interval graphs that provide the neighborhood of any vertex in O(d) time, where d is its degree. To that purpose, we introduce a new parameter, called linearity, that would solve the problem if bounded for the two classes. Surprisingly, we show that it is not. Nevertheless, we design representations with the desired property for the two classes, and we implement the Breadth-First Search algorithm in O(n) time for permutation graphs; thereby lowering the complexity of All Pairs Shortest Paths and Single Source Shortest Path problems for the class.
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References
Boldi, P., Vigna, S.: The webgraph framework I: compression techniques. In: WWW 2004, pp. 595–602. ACM, New York (2004)
Boldi, P., Vigna, S.: Codes for the world wide web. Internet Mathematics 2(4), 407–429 (2005)
Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: a Survey. SIAM Monographs on Discrete Mathematics and Applications (1999)
Gabow, H.N., Bentley, J.L., Tarjan, R.E.: Scaling and related techniques for geometry problems. In: STOC 1984, pp. 135–143 (1984)
Gavoille, C., Peleg, D.: The compactness of interval routing. SIAM Journal on Discrete Mathematics 12(4), 459–473 (1999)
Goldberg, P.W., Golumbic, M.C., Kaplan, H., Shamir, R.: Four strikes against physical mapping of DNA. Journal of Computational Biology 2(1), 139–152 (1995)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics, 2nd edn., vol. 57. Elsevier, Amsterdam (2004)
Harel, D.: A linear time algorithm for the lowest common ancestors problem (extended abstract). In: FOCS 1980, pp. 308–319 (1980)
Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestors. SIAM J. Comput. 13(2), 338–355 (1984)
Roberts, F.S.: Representations of Indifference Relations. PhD thesis, Stanford University (1968)
Spinrad, J.P.: Efficient graph representations. Fields Institute Monographs, vol. 19. American Mathematical Society (2003)
Turan, G.: On the succinct representation of graphs. Discr. Appl. Math. 8, 289–294 (1984)
Vuillemin, J.: A unifying look at data structures. Commun. ACM 23(4), 229–239 (1980)
Wang, R., Lau, F.C.M., Zhao, Y.: Hamiltonicity of regular graphs and blocks of consecutive ones in symmetric matrices. Discr. Appl. Math. 155(17), 2312–2320 (2007)
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Crespelle, C., Gambette, P. (2009). Efficient Neighborhood Encoding for Interval Graphs and Permutation Graphs and O(n) Breadth-First Search. In: Fiala, J., Kratochvíl, J., Miller, M. (eds) Combinatorial Algorithms. IWOCA 2009. Lecture Notes in Computer Science, vol 5874. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10217-2_17
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DOI: https://doi.org/10.1007/978-3-642-10217-2_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-10216-5
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