Abstract
A (1 + ε)–approximate distance oracle for a graph is a data structure that supports approximate point-to-point shortest-path-distance queries. The most relevant measures for a distance-oracle construction are: space, query time, and preprocessing time.
There are strong distance-oracle constructions known for planar graphs (Thorup, JACM’04) and, subsequently, minor-excluded graphs (Abraham and Gavoille, PODC’06). However, these require \(\Omega(\epsilon^{-1} n \lg n)\) space for n–node graphs.
In this paper, for planar graphs, bounded-genus graphs, and minor-excluded graphs we give distance-oracle constructions that require only O(n) space. The big O hides only a fixed constant, independent of ε and independent of genus or size of an excluded minor. The preprocessing times for our distance oracle are also faster than those for the previously known constructions. For planar graphs, the preprocessing time is \(O(n \lg^2 n)\). However, our constructions have slower query times. For planar graphs, the query time is \(O(\epsilon^{-2} \lg^2 n)\).
For all our linear-space results, we can in fact ensure, for any δ > 0, that the space required is only 1 + δ times the space required just to represent the graph itself.
An extended version can be found online [KKS11].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Acar, U.A., Blelloch, G.E., Harper, R., Vittes, J.L., Woo, S.L.M.: Dynamizing static algorithms, with applications to dynamic trees and history independence. In: Proceedings of the Fifteenth ACM-SIAM Symposium on Discrete Algorithms, pp. 531–540 (2004)
Abraham, I., Gavoille, C.: Object location using path separators. In: Proceedings of the Twenty-Fifth Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. 188–197 (2006); Details in LaBRI Research Report RR-1394-06
Alstrup, S., Holm, J., de Lichtenberg, K., Thorup, M.: Maintaining information in fully dynamic trees with top trees. ACM Transactions on Algorithms 1(2), 243–264 (2005)
Bartal, Y., Gottlieb, L.-A., Kopelowitz, T., Lewenstein, M., Roditty, L.: Fast, precise and dynamic distance queries. CoRR, abs/1008.1480 (2010) (to appear in SODA 2011)
Cabello, S.: Many distances in planar graphs. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1213–1220 (2006); a preprint of the journal version is available in the University of Ljubljana preprint series, vol. 47, p. 1089 (2009)
Cabello, S., Chambers, E.W.: Multiple source shortest paths in a genus g graph. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, New Orleans, Louisiana, USA, pp. 89–97 (2007)
Chen, D.Z., Xu, J.: Shortest path queries in planar graphs. In: Proceedings of the ACM Symposium on Theory of Computing (STOC), pp. 469–478 (2000)
Djidjev, H., Pantziou, G.E., Zaroliagis, C.D.: Improved algorithms for dynamic shortest paths. Algorithmica 28(4), 367–389 (2000)
Eppstein, D.: Dynamic generators of topologically embedded graphs. In: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 599–608 (2003)
Fakcharoenphol, J., Rao, S.: Planar graphs, negative weight edges, shortest paths, and near linear time. Journal of Computer and System Sciences 72(5), 868–889 (2006), announced at FOCS 2001
Frederickson, G.N.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM Journal on Computing 16(6), 1004–1022 (1987)
Frederickson, G.N.: A data structure for dynamically maintaining rooted trees. Journal of Algorithms 24, 37–65 (1997), announced at SODA 1993
Henzinger, M.R., Klein, P.N., Rao, S., Subramanian, S.: Faster shortest-path algorithms for planar graphs. Journal of Computer and System Sciences 55(1), 3–23 (1997), announced at STOC 1994
Har-Peled, S., Mendel, M.: Fast construction of nets in low dimensional metrics, and their applications. SIAM J. Comput. 35(5), 1148–1184 (2006), announced at SOCG 2005
Kowalik, L., Kurowski, M.: Oracles for bounded-length shortest paths in planar graphs. ACM Transactions on Algorithms 2(3), 335–363 (2006), announced at STOC 2003
Kawarabayashi, K., Klein, P.N., Sommer, C.: Linear-space approximate distance oracles for planar, bounded-genus, and minor-free graphs. CoRR, abs/1104.5214 (2011)
Klein, P.N.: Multiple-source shortest paths in planar graphs. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 146–155 (2005)
Klein, P.N., Mozes, S., Weimann, O.: Shortest paths in directed planar graphs with negative lengths: A linear-space O(nlog2n)-time algorithm. ACM Transactions on Algorithms 6(2) (2010), announced at SODA 2009
Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM Journal on Applied Mathematics 36(2), 177–189 (1979)
Mozes, S., Sommer, C.: Exact distance oracles for planar graphs. CoRR, abs/1011.5549 (2010)
Mozes, S., Wulff-Nilsen, C.: Shortest paths in planar graphs with real lengths in O(nlog2 n/loglogn) time. In: de Berg, M., Meyer, U. (eds.) ESA 2010. LNCS, vol. 6347, pp. 206–217. Springer, Heidelberg (2010)
Muller, L.F., Zachariasen, M.: Fast and compact oracles for approximate distances in planar graphs. In: Proceedings of the 15th Annual European Conference on Algorithms, pp. 657–668 (2007)
Nussbaum, Y.: Improved distance queries in planar graphs. CoRR, abs/1012.2825 (2010)
Patrascu, M., Roditty, L.: Distance oracles beyond the Thorup–Zwick bound. In: 51st Annual IEEE Symposium on Foundations of Computer Science, FOCS (2010)
Slivkins, A.: Distance estimation and object location via rings of neighbors. Distributed Computing 19(4), 313–333 (2007), announced at PODC 2005
Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. Journal of Computer and System Sciences 26(3), 362–391 (1983), announced at STOC 1981
Sommer, C., Verbin, E., Yu, W.: Distance oracles for sparse graphs. In: 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 703–712 (2009)
Talwar, K.: Bypassing the embedding: algorithms for low dimensional metrics. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), pp. 281–290 (2004)
Thorup, M.: Undirected single-source shortest paths with positive integer weights in linear time. Journal of the ACM 46(3), 362–394 (1999), announced at FOCS 1997
Thorup, M.: Floats, integers, and single source shortest paths. Journal of Algorithms 35(2), 189–201 (2000), announced at STACS 1998
Thorup, M.: Compact oracles for reachability and approximate distances in planar digraphs. Journal of the ACM 51(6), 993–1024 (2004), announced at FOCS 2001
Tarjan, R.E., Werneck, R.F.F.: Self-adjusting top trees. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 813–822 (2005)
Thorup, M., Zwick, U.: Approximate distance oracles. Journal of the ACM 52(1), 1–24 (2005), announced at STOC 2001
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kawarabayashi, Ki., Klein, P.N., Sommer, C. (2011). Linear-Space Approximate Distance Oracles for Planar, Bounded-Genus and Minor-Free Graphs. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-22006-7_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22005-0
Online ISBN: 978-3-642-22006-7
eBook Packages: Computer ScienceComputer Science (R0)