Skip to main content
SpringerLink
Log in
Book cover

International Symposium on Mathematical Foundations of Computer Science

MFCS 2008: Mathematical Foundations of Computer Science 2008 pp 68–82Cite as

Algebraic Graph Algorithms

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5162))

Abstract

The aim of this paper is to survey the results on dynamic algebraic algorithms, with main interest in matrix functions such as, determinant, inverse, rank and characteristic polynomial. First of all we summary the papers that in dynamic setup these problems can be solved faster than evaluating everything from scratch. The static complexity of these problem equals the matrix multiplication complexity, whereas the presented solutions work in subquadratic or quadratic (characteristic polynomial) time in the worst case. The dynamic matrix computations can be used to solve the following graphs problems in dynamic setup: computing transitive closure, computing shortest paths lengths, computing maximum matching size and computing vertex connectivity. For all of these problem the dynamic approach lead to the first known subquadratic algorithms. Astonishingly, the dynamic matrix algorithms can be used to obtain efficient static algorithms for the perfect matching problem as well. Using the O(n 2) algorithms for the dynamic matrix inverse, one can obtain a very simple randomized algorithm for computing perfect matchings in O(n 3) time. When the fast matrix multiplication is used, the complexity of this algorithm can be improved to O(n ω) time, where ω is the exponent of the best known matrix multiplication algorithm. Since ω< 2.38, this algorithm breaks through the O(n 2.5) barrier for the matching problem. The interplay between algebraic algorithms and graphs problems can be explored even further in order to obtain O(Wn ω) time algorithms for single source shortest paths problem and weighted bipartite matching problem.

This is a preview of subscription content, 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   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ausiello, G., Italiano, G.F., Marchetti Spaccamela, A., Nanni, U.: Incremental algorithms for minimal length paths. J. Algorithms 12(4), 615–638 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baswana, S., Hariharan, R., Sen, S.: Improved decremental algorithms for maintaining transitive closure and all-pairs shortest paths. In: Proceedings of the thiry-fourth annual ACM symposium on Theory of Computing, pp. 117–123. ACM Press, New York (2002)

    Chapter  Google Scholar 

  3. Bellman, R.: On a Routing Problem. Quarterly of Applied Mathematics 16(1), 87–90 (1958)

    MATH  MathSciNet  Google Scholar 

  4. Blum, N.: A new approach to maximum matching in general graphs. In: Proc.17th ICALP. LNCS, vol. 443, pp. 586–597. Springer, Heidelberg (1990)

    Google Scholar 

  5. Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.): ICALP 2006. LNCS, vol. 4051. Springer, Heidelberg (2006)

    Google Scholar 

  6. Cheriyan, J., Reif, J.H.: Directed s-t numberings, rubber bands, and testing digraph k-vertex connectivity. In: SODA 1992: Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms, pp. 335–344. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (1992)

    Google Scholar 

  7. Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. In: Proceedings of the nineteenth annual ACM conference on Theory of computing, pp. 1–6. ACM Press, New York (1987)

    Chapter  Google Scholar 

  8. Demetrescu, C., Italiano, G.F.: Fully Dynamic Transitive Closure: Breaking Through the O(n 2) Barrier. In: Proceedings of 41th annual IEEE Symposium on Foundations of Computer Science, pp. 381–389 (2000)

    Google Scholar 

  9. Demetrescu, C., Italiano, G.F.: Fully Dynamic All Pairs Shortest Paths with Real Edge Weights. In: Proceedings of 42th annual IEEE Symposium on Foundations of Computer Science, pp. 260–267 (2001)

    Google Scholar 

  10. Demetrescu, C., Italiano, G.F.: Improved Bounds and New Trade-Offs for Dynamic All Pairs Shortest Paths. In: Proceedings of the 29th International Colloquium on Automata, Languages and Programming, pp. 633–643. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  11. Demetrescu, C., Italiano, G.F.: A new approach to dynamic all pairs shortest paths. In: Proceedings of the thirty-fifth annual ACM Symposium on Theory of Computing, pp. 159–166. ACM Press, New York (2003)

    Chapter  Google Scholar 

  12. Diks, K., Sankowski, P.: Dynamic Plane Transitive Closure. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 594–604. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  13. Dinic, E.A., Kronrod, M.A.: An Algorithm for the Solution of the Assignment Problem. Soviet Math. Dokl. 10, 1324–1326 (1969)

    MATH  Google Scholar 

  14. Edmonds, J., Karp, R.M.: Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems. J. ACM 19(2), 248–264 (1972)

    Article  MATH  Google Scholar 

  15. Even, S., Gazit, H.: Updating Distances in Dynamic Graphs. Methods of Operations Research 49, 371–387 (1985)

    MATH  MathSciNet  Google Scholar 

  16. Fakcharoenphol, J.: Planar Graphs, Negative Weight Edges, Shortest Paths, and Near Linear Time. In: Proceedings of the 42nd IEEE symposium on Foundations of Computer Science, pp. 232–241. IEEE Computer Society Press, Los Alamitos (2001)

    Google Scholar 

  17. Frandsen, G.S., Hansen, J.P., Miltersen, P.B.: Lower Bounds for Dynamic Algebraic Problems. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  18. Frandsen, G.S., Frandsen, P.F.: Dynamic matrix rank. In: Bugliesi, et al. (eds.) [5], pp. 395–406

    Google Scholar 

  19. Frandsen, G.S., Sankowski, P.: Dynamic normal forms and dynamic characteristic polynomial. In: ICALP Proc. 35th ICALP. Springer, Heidelberg (2008)

    Google Scholar 

  20. Fredman, M.L.: The Complexity of Maintaining an Array and Computing Its Partial Sums. J. ACM 29(1), 250–260 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  21. Frigioni, D., Marchetti-Spaccamela, A., Nanni, U.: Semi-dynamic Algorithms for Maintaining Single Source Shortest Paths Trees. Algorithmica 22(3), 250–274 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Frigioni, D., Marchetti-Spaccamela, A., Nanni, U.: Fully Dynamic Algorithms for Maintaining Shortest Paths Trees. J. Algorithms 34(2), 251–281 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Gabow, H.N.: Using expander graphs to find vertex connectivity. In: FOCS 2000: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, p. 410. IEEE Computer Society, Washington, DC, USA (2000)

    Chapter  Google Scholar 

  24. Gabow, H.N.: Scaling Algorithms for Network Problems. J. Comput. Syst. Sci. 31(2), 148–168 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  25. Gabow, H.N., Tarjan, R.E.: Faster Scaling Algorithms for Network Problems. SIAM J. Comput. 18(5), 1013–1036 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  26. Gabow, H.N., Tarjan, R.E.: Faster scaling algorithms for general graph matching problems. J. ACM 38(4), 815–853 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  27. Andrew, V.: Scaling algorithms for the shortest paths problem. In: SODA 1993: Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms, pp. 222–231.Society for Industrial and Applied Mathematics (1993)

    Google Scholar 

  28. Nicholas, J.A.: Algebraic structures and algorithms for matching and matroid problems. In: FOCS 2006: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 531–542. IEEE Computer Society, Washington, DC, USA (2006)

    Google Scholar 

  29. Henzinger, M.R., King, V.: Fully Dynamic Biconnectivity and Transitive Closure. In: Proceedings 36th annual IEEE Symposiumon Foundations of Computer Science, pp. 664–672 (1995)

    Google Scholar 

  30. Henzinger, M.R., Klein, P., Rao, S., Subramanian, S.: Faster Shortest-path Algorithms for Planar Graphs. J. Comput. Syst. Sci. 55(1), 3–23 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  31. Huang, X., Pan, V.Y.: Fast Rectangular Matrix Multiplication and Applications. Journal of complexity 14(2), 257–299 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  32. Ibaraki, T., Katoh, N.: On-line Computation of Transitive Closure for Graphs. Inform. Proc. Lett. 16, 95–97 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  33. Iri, M.: A new method for solving transportation-network problems. Journal of the Operations Research Society of Japan 3, 27–87 (1960)

    Google Scholar 

  34. Italiano, G.F.: Amortized Efficiency of a Path Retrieval Data Structure. Theor. Comput. Sci. 48(2-3), 273–281 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  35. Ford Jr., L.R.: Network Flow Theory. Paper P-923, The RAND Corperation, Santa Moncia, California (August 1956)

    Google Scholar 

  36. Kao, M.-Y., Lam, T.W., Sung, W.-K., Ting, H.-F.: A decomposition theorem for maximum weight bipartite matchings with applications to evolutionary trees. In: Proceedings of the 7th Annual European Symposium on Algorithms, pp. 438–449 (1999)

    Google Scholar 

  37. Karp, R.M., Upfal, E., Wigderson, A.: Constructing a perfect matching is in random nc. Combinatorica 6(1), 35–48 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  38. Khanna, S., Motwani, R., Wilson, R.H.: On Certificates and Lookahead on Dynamic Graph Problems. In: Proceedings 7th annual ACM-SIAM Symposiumon on Discrete Algorithms, pp. 222–231 (1996)

    Google Scholar 

  39. King, V.: Fully Dynamic Algorithms for Maintaining All-Pairs Shortest Paths and Transitive Closure in Digraphs. In: Proceedings of 40th annual IEEE Symposium on Foundations of Computer Science, pp. 81–91 (1999)

    Google Scholar 

  40. King, V., Sagert, G.: A Fully Dynamic Algorithm for Maintaining the Transitive Closure. In: Proceedings of the thirty-first annual ACM Symposium on Theory of Computing, pp. 492–498. ACM Press, New York (1999)

    Chapter  Google Scholar 

  41. Kuhn, H.W.: The Hungarian Method for the Assignment Problem. Naval Research Logistics Quarterly 2, 83–97 (1955)

    Article  MathSciNet  Google Scholar 

  42. La Poutré, J.A., van Leeuwen, J.: Maintenance of Transitive Closure and Transitive Reduction of Graphs. In: Proc. Workshop on Graph-Theoretic Concepts in Computer Science. LNCS, vol. 314, pp. 106–120. Springer, Berlin (1988)

    Google Scholar 

  43. Lipton, R.J., Tarjan, R.E.: Applications of a planar separator theorem. SIAM J. Comput. 9(3), 615–627 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  44. Loubal, P.: A Network Evaluation Procedure. Highway Research Record 205, 96–109 (1967)

    Google Scholar 

  45. Micali, S., Vazirani, V.V.: An \(O(\sqrt{|V|}|E|)\) algorithm for finding maximum matching in general graphs. In: Proceedings of the twenty first annual IEEE Symposium on Foundations of Computer Science, pp. 17–27 (1980)

    Google Scholar 

  46. Miller, G.L., Naor, J.: Flow in planar graphs with multiple sources and sinks. SIAM J. Comput. 24(5), 1002–1017 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  47. Moore, E.F.: The Shortest Path Through a Maze. In: Proceedings of the International Symposium on the Theory of Switching, pp. 285–292. Harvard University Press (1959)

    Google Scholar 

  48. Mucha, M., Sankowski, P.: Maximum Matchings in Planar Graphs via Gaussian Elimination. In: Proceedings of 12th annual European Symposium on Algorithms, pp. 532–543 (2004)

    Google Scholar 

  49. Mucha, M., Sankowski, P.: Maximum Matchings via Gaussian Elimination. In: Proceedings of the 45th annual IEEE Symposium on Foundations of Computer Science, pp. 248–255 (2004)

    Google Scholar 

  50. Mucha, M., Sankowski, P.: Maximum matchings via gaussian elimination. In: Proceedings of the 45th annual IEEE Symposium on Foundations of Computer Science, pp. 248–255 (2004)

    Google Scholar 

  51. Mucha, M., Sankowski, P.: Fast dynamic transitive closure with lookahead. Algorithmica (2008)

    Google Scholar 

  52. Mulmuley, K., Vazirani, U.V., Vazirani, V.V.: Matching is as easy as matrix inversion. In: STOC 1987: Proceedings of the nineteenth annual ACM conference on Theory of computing, pp. 345–354. ACM Press, New York (1987)

    Chapter  Google Scholar 

  53. Munkres, J.: Algorithms for the Assignment and Transportation Problems. Journal of SIAM 5(1), 32–38 (1957)

    MATH  MathSciNet  Google Scholar 

  54. Murchland, J.: The Effect of Increasing or Decreasing the Length of a Single Arc on All Shortest Distances in a Graph. Technical report, LBS-TNT-26 (1967)

    Google Scholar 

  55. Lovász, L., Linial, N., Wigderson, A.: Rubber bands, convex embeddings and graph connectivity. Combinatorica 8, 91–102 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  56. Rabin, M.O., Vazirani, V.V.: Maximum matchings in general graphs through randomization. Journal of Algorithms 10, 557–567 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  57. Ramalingam, G., Reps, T.: An Incremental Algorithm for a Generalization of the Shortest-path Problem. J. Algorithms 21(2), 267–305 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  58. Ramalingam, G., Reps, T.: On the Computational Complexity of Dynamic Graph Problems. Theor. Comput. Sci. 158(1-2), 233–277 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  59. Reif, J.H., Tate, S.R.: On Dynamic Algorithms for Algebraic Problems. J. Algorithms 22(2), 347–371 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  60. Rodionov, V.: The Parametric Problem of Shortest Distances. U.S.S.R. Computational Math. and Math. Phys. 8(5), 336–343 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  61. Roditty, L.: A Faster and Simpler Fully Dynamic Transitive Closure. In: Proceedings of the fourteenth annual ACM-SIAM Symposium on Discrete Algorithms, pp. 404–412. Society for Industrial and Applied Mathematics (2003)

    Google Scholar 

  62. Roditty, L., Zwick, U.: Improved Dynamic Reachability Algorithms for Directed Graphs. In: Proceedings of the 43rd Symposium on Foundations of Computer Science, p. 679. IEEE Computer Society, Los Alamitos (2002)

    Google Scholar 

  63. Roditty, L., Zwick, U.: A Fully Dynamic Reachability Algorithm for Directed Graphs with an Almost Linear Update Time. In: Proceeding of the 36th annual ACM Symposium on Theory of Computing, pp. 184–191. ACM Press, New York (2004)

    Google Scholar 

  64. Rohnert, H.: A Dynamization of the All Pairs Least Cost Path Problem. In: Proceedings of the 2nd Symposium of Theoretical Aspects of Computer Science, pp. 279–286. Springer, Heidelberg (1985)

    Google Scholar 

  65. Sankowski, P.: Dynamic Transitive Closure via Dynamic Matrix Inverse. In: Proceedings of the 45th annual IEEE Symposium on Foundations of Computer Science, pp. 509–517 (2004)

    Google Scholar 

  66. Sankowski, P.: Processor efficient parallel matching. In: SPAA 2005: Proceedings of the seventeenth annual ACM symposium on Parallelism in algorithms and architectures. ACM Press, New York, NY, USA (2005)

    Google Scholar 

  67. Sankowski, P.: Shortest Paths in Matrix Multiplication Time. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 770–778. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  68. Sankowski, P.: Subquadratic Algorithm for Dynamic Shortest Distances. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 461–470. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  69. Sankowski, P.: Weighted bipartite matching in matrix multiplication time. In: Bugliesi, et al. (eds.) [5], pp. 274–285

    Google Scholar 

  70. Sankowski, P.: Faster dynamic matchings and vertex connectivity. In: SODA 2007: Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms, pp. 118–126. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2007)

    Google Scholar 

  71. Sherman, J., Morrison, W.J.: Adjustment of an inverse matrix corresponding to changes in the elements of a given column or a given row of the original matrix. Ann. Math. Statist. 20(5), 621 (1949)

    Google Scholar 

  72. Sherman, J., Morrison, W.J.: Adjustment of an inverse matrix corresponding to a change in one element of a given matrix. Ann. Math. Statist. 21(1), 124–127 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  73. Shimbel, A.: Structure in Communication Nets. In: Proceedings of the Symposium on Information Networks, pp. 199–203. Polytechnic Press of the Polytechnic Institute of Brooklyn, Brooklyn (1955)

    Google Scholar 

  74. Storjohann, A.: High-order lifting and integrality certification. J. Symb. Comput. 36(3-4), 613–648 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  75. Subramanian, S.: A fully dynamic data structure for reachability in planar digraphs. In: ESA 1993: Proceedings of the First Annual European Symposium on Algorithms, pp. 372–383. Springer, London, UK (1993)

    Google Scholar 

  76. Thorup, M.: Fully-dynamic all-pairs shortest paths: Faster and allowing negative cycles. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 384–396. Springer, Heidelberg (2004)

    Google Scholar 

  77. Thorup, M.: Worst-case update times for fully-dynamic all-pairs shortest paths. In: STOC 2005: Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, pp. 112–119. ACM, New York, NY, USA (2005)

    Chapter  Google Scholar 

  78. Yellin, D.M.: Speeding up Dynamic Transitive Closure for Bounded Degree Graphs. Acta Informatica 30, 369–384 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  79. Yuster, R., Zwick, U.: Answering distance queries in directed graphs using fast matrix multiplication. In: FOCS, pp. 389–396. IEEE Computer Society, Los Alamitos (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Warsaw University and ETH Zürich,  

    Piotr Sankowski

Authors
  1. Piotr Sankowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Edward Ochmański Jerzy Tyszkiewicz

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Sankowski, P. (2008). Algebraic Graph Algorithms. In: Ochmański, E., Tyszkiewicz, J. (eds) Mathematical Foundations of Computer Science 2008. MFCS 2008. Lecture Notes in Computer Science, vol 5162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85238-4_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-85238-4_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85237-7

  • Online ISBN: 978-3-540-85238-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation