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Compact oracles for reachability and approximate distances in planar digraphs

Published: 01 November 2004 Publication History

Abstract

It is shown that a planar digraph can be preprocessed in near-linear time, producing a near-linear space oracle that can answer reachability queries in constant time. The oracle can be distributed as an O(log n) space label for each vertex and then we can determine if one vertex can reach another considering their two labels only.The approach generalizes to give a near-linear space approximate distances oracle for a weighted planar digraph. With weights drawn from {0, …, N}, it approximates distances within a factor (1 + ε) in O(log log (nN) + 1/ε) time. Our scheme can be extended to find and route along correspondingly short dipaths.

References

[1]
Arikati, S., Chen, D., Chew, L., Das, G., Smid, M., and Zaroliagis, C. 1996. Planar spanners and approximate shortest path queries among obstacles in the plane. In Proceedings of the 4th European Symposium on Algorithms (Barcelona, Spain). Lecture Notes in Computer Science, vol. 1136. Springer-Verlag, New York, 514--528.]]
[2]
Chen, D. 1995. On the all-pairs Euclidian shortest path problem. In Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms (San Francisco, Calif.). ACM, New York, 292--301.]]
[3]
Chen, D., and Xu, J. 2000. Shortest path queries in planar graphs. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (Portland, Ore.). ACM, New York, 469--478.]]
[4]
Djidjev, H. 1996. Efficient algorithms for shortest path queries in planar digraphs. In Proceedings of the 22nd International Workshop on Graph-Theoretic Concepts in Computer Science (Como, Italy). Lecture Notes in Computer Science, vol. 1197. Springer-Verlag, New York, 151--165.]]
[5]
Djidjev, H., Panziou, G., and Zaroliagis, C. 1991. Computing shortest paths and distances in planar graphs. In Proceedings of the 18th International Colloquium on Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 1991. Springer-Verlag, New York, 327--339.]]
[6]
Djidjev, H., Panziou, G., and Zaroliagis, C. 1995. Fast algorithms for maintaining shortest paths in outerplanar and planar digraphs. In Proceeding of 10th International Symposium on Fundamentals of Computation Theory. Lecture Notes in Computer Science, vol. 965. Springer-Verlag, New York, 191--200.]]
[7]
Frederickson, G. 1987. Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput. 16, 1004--1022.]]
[8]
Gavoille, C., Peleg, D., Pérennes, S., and Raz, R. 2001. Distance labeling in graphs. In Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (Washington, D.C.). ACM, New York, 210--219.]]
[9]
Gilbert, J. R., Hutchinson, J. P., and Tarjan, R. 1984. A separator theorem for graphs of bounded genus. J. Algorithms 5, 391--407.]]
[10]
Gupta, A., Kumar, A., and Rastogi, R. 2001. Traveling with a pez dispenser (or, routing issues in MPLS). In Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Alamitos, Calif., 148--157.]]
[11]
Gupta, A., Kumar, A., and Thorup, M. 2003. Tree based MPLS routing. In Proceedings of the 15th ACM Symposium on Parallel Algorithms. ACM, New York, 193--199.]]
[12]
Hagerup, T., Miltersen, P., and Pagh, R. 2001. Deterministic dictionaries. J. Algorithms 41, 1, 69--85.]]
[13]
Harel, D., and Tarjan, R. 1984. Fast algorithms for finding nearest common ancestor. SIAM J. Comput. 13, 338--355.]]
[14]
Henzinger, M., Klein, P., Rao, S., and Subramanian, S. 1997. Faster shortest-path algorithms for planar graphs. J. Comput. Syst. Sci. 55, 3--23.]]
[15]
Indyk, P. 1999. Sublinear time algorithms for metric space problems. In Proceedings of the 31th Annual ACM Symposium on Theory of Computing (Atlanta, Ga.). ACM, New York, 428--434.]]
[16]
Kernighan, B., and Ritchie, D. 1988. The C Programming Language, Second ed. Prentice-Hall.]]
[17]
Klein, P. 2002. Preprocessing an undirected planar network to enable fast approximate distance queries. In Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms (San Francisco, Calif.). ACM, New York, 820--827.]]
[18]
Klein, P., and Subramanian, S. 1998. A fully dynamic approximation scheme for shortest path problems in planar graphs. Algorithmica 23, 235--249.]]
[19]
Lipton, R., and Tarjan, R. 1979. A separator theorem for planar graphs. SIAM J. Appl. Math. 36, 177--189.]]
[20]
Peleg, D. 2000a. Informative labeling schemes for graphs. In Proceedings of the 25th Mathematical Foundations of Computer Science (Bratislava, Slovak Republik). Lecture Notes in Computer Science, vol. 1893. Springer-Verlag, New York, 579--588.]]
[21]
Peleg, D. 2000b. Proximity-preserving labeling schemes. J. Graph Theory 33, 167--176.]]
[22]
Robertson, N., and Seymour, P. S. 1986. Graph minors V: Excluding a planar graph. J. Combin. Theory, Ser. B 41, 1, 92--114.]]
[23]
Robertson, N., and Seymour, P. S. 2003. Graph minors XVI: Excluding a non-planar graph. J. Combin. Theory, Ser. B 89, 1, 43--76.]]
[24]
Santoro, N., and Khatib, R. 1985. Labelling and implicit routing in networks. Comput. J. 28, 1, 5--8.]]
[25]
Subramanian, S. 1993. A fully dynamic data structure for reachability in planar digraphs. In Proceedings of the 1st European Symposium on Algorithms (Bad Honnef, Germany). Lecture Notes in Computer Science, vol. 726. Springer-Verlag, New York, 372--383.]]
[26]
Tamassia, R., and Tollis, I. 1993. Dynamic reachability in planar digraphs with one source and one sink. Theoret. Comput. Sci. 119, 331--343.]]
[27]
Thorup, M. 1995. Shortcutting planar digraphs. Combin., Prob. Comput. 4, 287--315.]]
[28]
Thorup, M. 1999. Undirected single-source shortest paths with positive integer weights in linear time. J. ACM 46, 362--394.]]
[29]
Thorup, M. 2000. On RAM priority queues. SIAM J. Comput. 30, 1, 86--109.]]
[30]
Thorup, M., and Zwick, U. 2001a. Approximate distance oracles. In Proceedings of the 33th Annual ACM Symposium on Theory of Computing (Crete, Greece). ACM, New York, 183--192.]]
[31]
Thorup, M., and Zwick, U. 2001b. Compact routing schemes. In Proceedings of the 13th Annual ACM Symposium on Parallel Algorithms and Architectures (Crete, Greece). ACM, New York, 1--10.]]
[32]
van Leeuwen, J., and Tan, R. 1986. Computer networks with compact routing tables. In The book of L, G. Rozenberg and A. Salomaa, Eds. Springer-Verlag, New York, 259--273.]]
[33]
Williams, J. 1964. Heapsort. Commun. ACM 7, 5, 347--348.]]

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Published In

cover image Journal of the ACM
Journal of the ACM  Volume 51, Issue 6
November 2004
191 pages
ISSN:0004-5411
EISSN:1557-735X
DOI:10.1145/1039488
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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 November 2004
Published in JACM Volume 51, Issue 6

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  1. Planar graphs
  2. reachability and shortest paths oracles

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