Advertisement

Computing shortest paths and distances in planar graphs

  • Hristo N. Djidjev
  • Grammati E. Pantziou
  • Christos D. Zaroliagis
Graph Algorithms (Session 8)
Part of the Lecture Notes in Computer Science book series (LNCS, volume 510)

Abstract

We provide here efficient sequential and parallel solutions to the following problem: given a planar digraph G (with real edge weights but no negative cycles) for preprocessing, answer on-line queries requesting the shortest distance (or path) between any two vertices in G. Our algorithms for preprocessing need O(n log n + q2) space and O(n log n + q2) sequential time. (Here q is the cardinality of a set of faces of a planar embedding of G that cover all vertices.)A parallel implementation on a CREW PRAM needs also O(n log n + q2) space and runs in O(log2n) time using O(n + M(q)) processors (where M(q) is the number of processors required to multiply two q × q matrices in O(log q) time), provided that the q faces are given by the input.This enables us to achieve O(log n) time using a single processor for a “distance” query, or O(L + log n) time for a “path” query (where L is the length of the path). Note that this is a considerable improvement over previous results in the case where q = o(n). Our techniques are based on the hammock decomposition of a planar digraph and the use of separators for computing quickly internal distances in the graph. Several other results are achieved. For outerplanar graphs, our algorithms preprocess the graph in O(n logn) space and run either in O(n log n) sequential time, or in O(log2n) time using O(n) processors on a CREW PRAM. A “distance” query can be answered in O(log n) time using a single processor. A “path” query is answered in O(L + log n) time. An optimal solution is given in the case of trees. We achieve O(1) time per “distance” query andwe need O(n) sequential time, or O(log n) time and O(n/log n) processors (on an EREW PRAM) for preprocessing. A “path” query is answered in O(L) time.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G. Ausiello, G.F. Italiano, A.M. Spaccamela, U. Nanni, “Incremental algorithms for minimal length paths”, Proc. of ACM-SIAM SODA, 1990, pp.12–21.Google Scholar
  2. [2]
    D. Beinstock, C.L. Monma, “On the complexity of covering faces by vertices in a planar graph”, SIAM J. Comp., Vol.17, No.1, Feb. 1988, pp.53–76.Google Scholar
  3. [3]
    B. Berger, J. Rompel, P.W. Shor, “Efficient NC-algorithms for set cover with applications to learning and geometry”, Proc. 30th IEEE Symp. on FOCS, 1989, pp.54–59.Google Scholar
  4. [4]
    H. Djidjev, G. Pantziou, C. Zaroliagis, “Computing Shortest Paths and Distances in Planar Graphs”, CTI TR-90.10.26, Computer Technology Institute, Patras, October 1990.Google Scholar
  5. [5]
    G.N. Frederickson, “Fast Algorithms for Shortest Paths in Planar Graphs with Applications”, SIAM J. Comp., Vol.16, 1987, pp.1004–1022.Google Scholar
  6. [6]
    G.N. Frederickson, “Planar Graph Decomposition and All Pairs Shortest Paths”, TR-89-015, ICSI, Berkeley, March 1989. A preliminary version was appeared as “A new approach to all pairs shortest paths in planar graphs”, Proc. 19th ACM STOC, New York City, May 1987, pp.19–28.Google Scholar
  7. [7]
    G.N. Frederickson “Using Cellular Graph Embeddings in Solving All Pairs Shortest Paths Problems”, CSD-TR-897, Purdue University, August 1989. A preliminary version was appeared in Proc. 30th IEEE Symp. on FOCS, 1989, pp.448–453.Google Scholar
  8. [8]
    G.N. Frederickson, R. Janardan, “Designing Networks with Compact Routing Tables”, Algorithmica, 3 (1988), pp.171–190.Google Scholar
  9. [9]
    M.L. Fredman, R.E. Tarjan, “Fibonacci heaps and their use in improved network optimization algorithms”, J. ACM, 34(1987), pp.596–615.Google Scholar
  10. [10]
    D. Knuth, “The Art of Computer Programming”, Vol.1, Fundamental Algorithms, 2nd ed. Addison-Wesley, 1973.Google Scholar
  11. [11]
    A. Lingas, “Efficient parallel algorithms for path problems in planar directed graphs”, Proc. of SIGAL90, LNCS, Vol.450, pp. 447–457.Google Scholar
  12. [12]
    G. Pantziou, P. Spirakis, C. Zaroliagis, “Efficient Parallel Algorithms for Shortest Paths in Planar Graphs”, CTI TR-90.01.02, Computer Technology Institute, Patras, September 1990 (revised version). A preliminary version has appeared as Proc. of the 2nd Scand. Workshop on Algorithm Theory (SWAT90), Bergen, Norway, 11–14 July, 1990, LNCS, Vol. 447, pp.288–300, Springer-Verlag.Google Scholar
  13. [13]
    B. Schieber, U. Vishkin, “On finding lowest common ancestors: simplification and parallelization”, Proc. 3rd AWOC88, Corfu, Greece, July 1988, LNCS Vol. 319 (ed. J.H. Reif), pp.111–123, Springer-Verlag.Google Scholar
  14. [14]
    R.E. Tarjan, U. Vishkin, “An efficient parallel biconnectivity algorithm”, SIAM J. Comp., 14 (1985), pp. 862–874.Google Scholar
  15. [15]
    J. van Leeuwen, R.B. Tan, “Computer Networks with compact routing tables”, in The Book of L, G. Rozenberg and A. Salomaa (eds.), Springer-Verlag, New York (1986), pp.259–273.Google Scholar
  16. [16]
    J.C. Wyllie, “The Complexity of Parallel Computation”, PhD Thesis, TR 79-387, Dept of Computer Science, Cornell University, Ithaca, NY, 1979.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Hristo N. Djidjev
    • 1
  • Grammati E. Pantziou
    • 2
    • 3
  • Christos D. Zaroliagis
    • 2
    • 3
  1. 1.Center of Informatics and Computer TechnologyBulgarian Academy of SciencesSofiaBulgaria
  2. 2.Computer Technology InstitutePatrasGreece
  3. 3.Computer Sc and Eng DeptUniversity of PatrasGreece

Personalised recommendations