Skip to main content
SpringerLink
Log in

A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs

  • Conference paper
Book cover SOFSEM 2012: Theory and Practice of Computer Science (SOFSEM 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7147))

Abstract

We consider the problem of computing all-pairs shortest paths in a directed graph with non-negative real weights assigned to vertices.

For an n×n 0 − 1 matrix C, let K C be the complete weighted graph on the rows of C where the weight of an edge between two rows is equal to their Hamming distance. Let MWT(C) be the weight of a minimum weight spanning tree of K C .

We show that the all-pairs shortest path problem for a directed graph G on n vertices with non-negative real weights and adjacency matrix A G can be solved by a combinatorial randomized algorithm in time

$$\widetilde{O}(n^{2}\sqrt{n + \min\{MWT(A_G), MWT(A_G^t)\}})$$

As a corollary, we conclude that the transitive closure of a directed graph G can be computed by a combinatorial randomized algorithm in the aforementioned time.

We also conclude that the all-pairs shortest path problem for vertex-weighted uniform disk graphs induced by point sets of bounded density within a unit square can be solved in time \(\widetilde{O}(n^{2.75})\).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974)

    MATH  Google Scholar 

  2. Alon, N., Naor, M.: Derandomization, Witnesses for Boolean Matrix Multiplication and Construction of Perfect hash functions. Algorithmica 16, 434–449 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alon, N., Galil, Z., Margalit, O.: On the exponent of all pairs shortest path problem. J. Comput. System Sci. 54, 25–51 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bansal, N., Williams, R.: Regularity Lemmas and Combinatorial Algorithms. In: Proc. of 50th IEEE Symposium on Foundations on Computer Science, Atlanta (2009)

    Google Scholar 

  5. Björklund, A., Lingas, A.: Fast Boolean Matrix Multiplication for Highly Clustered Data. In: Dehne, F., Sack, J.-R., Tamassia, R. (eds.) WADS 2001. LNCS, vol. 2125, pp. 258–263. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  6. Borodin, A., Ostrovsky, R., Rabani, Y.: Subquadratic Approximation Algorithms For Clustering Problems in High Dimensional Spaces. In: Proceedings of the 31st ACM Symposium on Theory of Computing (1999)

    Google Scholar 

  7. Cameron, P.J.: Combinatorics. Cambridge University Press (1994)

    Google Scholar 

  8. Chan, T.M.: More algorithms for all-pairs shortest paths in weighted graphs. SIAM J. Comput. 39(5), 2075–2089; preliminary version in proc. STOC 2007, pp. 590–598 (2007)

    Google Scholar 

  9. Chan, T.M.: All-pairs shortest paths with real weights in O(n 3/logn) time. Algorithmica 41, 330–337 (2008)

    Article  Google Scholar 

  10. Coppersmith, D., Winograd, S.: Matrix Multiplication via Arithmetic Progressions. J. of Symbolic Computation 9, 251–280 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fürer, M., Kasiviswanathan, S.P.: Approximate Distance Queries in Disk Graphs. In: Erlebach, T., Kaklamanis, C. (eds.) WAOA 2006. LNCS, vol. 4368, pp. 174–187. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  12. Galil, Z., Margalit, O.: Witnesses for Boolean Matrix Multiplication and Shortest Paths. Journal of Complexity, 417–426 (1993)

    Google Scholar 

  13. Gąsieniec, L., Lingas, A.: An Improved Bound on Boolean Matrix Multiplication for Highly Clustered Data. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 329–339. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  14. Huang, X., Pan, V.Y.: Fast rectangular matrix multiplications and applications. Journal of Complexity 14(2), 257–299 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Indyk, P.: High-dimensional computational geometry. PhD dissertation, Standford University (2000)

    Google Scholar 

  16. Indyk, P., Motwani, R.: Approximate Nearest Neighbors: Towards Removing the Curse of Dimensionality. In: Proceedings of the 30th ACM Symposium on Theory of Computing (1998)

    Google Scholar 

  17. Indyk, P., Schmidt, S.E., Thorup, M.: On reducing approximate mst to closest pair problems in high dimensions (1999) (manuscript)

    Google Scholar 

  18. Karp, R.M., Steele, J.M.: Probabilistic analysis of heuristics. In: Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B. (eds.) The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, ch. 6, pp. 181–205. John Wiley & Sons Ltd. (1985)

    Google Scholar 

  19. Kushilevitz, E., Ostrovsky, E., Rabani, Y.: Efficient search for approximate nearest neighbor in high dimensional spaces. SIAM J. Comput. 30(2), 457–474; Preliminary version in Proc. 30th STOC (1989)

    Google Scholar 

  20. Lingas, A.: A Geometric Approach to Boolean Matrix Multiplication. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 501–510. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  21. Munro, J.I.: Efficient determination of the transitive closure of a directed graph. Information Processing Letters 1(2), 56–58 (1971)

    Article  MATH  Google Scholar 

  22. Rytter, W.: Fast recognition of pushdown automaton and context-free languages. Information and Control 67(1-3), 12–22 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  23. Seidel, R.: On the All-Pairs-Shortest-Path Problem. In: Proc. 24th ACM STOC, pp. 745–749 (1992)

    Google Scholar 

  24. Vassilevska, V., Williams, R.: Subcubic Equivalences Between Path, Matrix, and Triangle Problems. In: Proceedings 51st Annual IEEE Symposium on Foundations of Computer Science, FOCS (2010)

    Google Scholar 

  25. Yuster, R.: Efficient algorithms on sets of permutations, dominance, and real-weighted APSP. In: Proc. of the 20th ACM-SIAM Symposium on Discrete Algorithms, pp. 950–957 (2009)

    Google Scholar 

  26. Zwick, U.: All pairs shortest paths using bridging rectangular matrix multiplication. Journal of the ACM 49(3), 289–317 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zwick, U.: Exact and Approximate Distances in Graphs - A survey. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 33–48. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Lund University, 22100, Lund, Sweden

    Andrzej Lingas

  2. The Centre for Mathematical Sciences, Lund University, 22100, Lund, Sweden

    Dzmitry Sledneu

Authors
  1. Andrzej Lingas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dzmitry Sledneu
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Informatics and Information Technologies, Institute of Informatics and Software Engineering, Slovak University of Technology in Bratislava, Ilkovičova 3, 842 16, Bratislava 4, Slovakia

    Mária Bieliková

  2. Dept. of Intelligent Systems and Business Informatics, Alpen-Adria-Universität Klagenfurt, Universitätsstr. 65-57, 9020, Klagenfurt, Austria

    Gerhard Friedrich

  3. Department of Computer Science, University of Oxford, UK

    Georg Gottlob

  4. Security Engineering Group, Technische Universität Darmstadt, Hochschulstr. 10, 64289, Darmstadt, Germany

    Stefan Katzenbeisser

  5. University of Illinois at Chicago, Dept. of Math., Stat. and Comp. Sci, 851 S. Morgan Street, Chicago, IL 60607-7045, USA; and University of Szeged, Research Group on Artificial Intelligence of the Hungarian Academy of Sciences, 6701 Szeged, Postafiók, Hungary

    György Turán

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Lingas, A., Sledneu, D. (2012). A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs. In: Bieliková, M., Friedrich, G., Gottlob, G., Katzenbeisser, S., Turán, G. (eds) SOFSEM 2012: Theory and Practice of Computer Science. SOFSEM 2012. Lecture Notes in Computer Science, vol 7147. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27660-6_31

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-27660-6_31

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-27659-0

  • Online ISBN: 978-3-642-27660-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation