Efficient algorithms for solving systems of linear equations and path problems

Extended abstract
  • Venkatesh Radhakrishnan
  • Harry B. HuntIII
  • Richard E. Stearns
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 577)


Efficient algorithms are presented for solving systems of linear equations defined on and for solving path problems [11] for treewidth k graphs [20] and for α-near-planar graphs [22]. These algorithms include the following:
  1. 1.

    O(nk2) and O(n3/2) time algorithms for solving a system of linear equations and for solving the single source shortest path problem,

  2. 2.

    O(n2k) and O(n2log n) time algorithms for computing A−1 where A is an n×n matrix over a field or for computing A* where A is an n×n matrix over a closed semiring, and

  3. 3.

    O(n2k) and O(n2log n) time algorithms for the all pairs shortest path problems.


One corollary of these results is that the single source and all pairs shortest path problems are solvable in O(n) and O(n2) steps, respectively, for any of the decomposable graph classes in [5].

Key words

Algorithms and data structures Mathematics of computation Systems of Linear Equations Path Problems Gaussian Elimination 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A.Arnborg, D.G.Corneil and A.Proskuworski, “Complexity of finding embeddings in a k tree”, SIAM J.Alg. and Discr.Methods 8, 1987, pp.277–284.Google Scholar
  2. [2]
    A.V.Aho, J.E.Hopcroft, J.D.Ullman, “The Design and Analysis of Computer Algorithms”, Addison-Wesley, 1974.Google Scholar
  3. [3]
    A.Arnborg, J.Lagergren, D.Seese,“Problems easy for decomposable graphs”,Proceedings of ICALP 88, Springer-Verlag LNCS 317 (1988), 38–51Google Scholar
  4. [4]
    R.C.Backhouse and B.A.Carre,“Regular algebra applied to pathfinding problems”, J. Inst. Math. Appl., 15,1974,pp.161–186.Google Scholar
  5. [5]
    M.W.Bern, E.L.Lawler and A.L.Wong, “Linear-time Computation of Optimal Subgraphs of Decomposable Graphs”,J. Alg. 8(1987),pp.216–235.Google Scholar
  6. [6]
    H.L.Bodlaender, “Dynamic programming on graphs with bounded tree-width”, RUU-CS-87-22, Proceedings of ICALP 88, Springer-Verlag LNCS 317(1988),105–118.Google Scholar
  7. [7]
    J.R.Bunch and D.J.Rose,“Partitioning, tearing, and modification of sparse linear systems”,J. Math. Anal. Appl. 48(1974),pp.574–593.Google Scholar
  8. [8]
    D.E.Comer,“Internetworking with TCP/IP”, vol.I, Prentice-Hall,1991.Google Scholar
  9. [9]
    G.Dahlquist and A.Björck,“Numerical methods”,Prentice-Hall,1974.Google Scholar
  10. [10]
    M.R.Garey and D.S.Johnson, “Computers and Intractability: A Guide to the Theory of NP-Completeness”, W.H.Freeman and Company, 1979.Google Scholar
  11. [11]
    M.Gondran and M.Minoux,“Graphs and Algorithms”,John Wiley,1984.Google Scholar
  12. [12]
    Y.Gurevich, L.Stockmeyer, and U.Vishkin, “Solving NP-hard problems on graphs that are almost trees and an application to facility location problems”,J. ACM, vol.31,pp.459–473,1984.Google Scholar
  13. [13]
    R.Hassin and A.Tamir, “Efficient algorithms for optimization and selection on series-parallel graphs”, SIAM J. Alg. Disc. Meth., vol 7, pp.379–389, 1986.Google Scholar
  14. [14]
    I.N.Herstein and D.J.Winter,“Matrix Theory and Linear Algebra”,Macmillan,1988.Google Scholar
  15. [15]
    D.J.Lehmann,“Algebraic structures for transitive closure”, Theor. Comp. Sc., vol. 4, pp.59–76, 1977.Google Scholar
  16. [16]
    R.L.Lipton and R.E.Tarjan, “Applications of a planar separator theorem”, SICOMP, vol.9, pp.615–629, 1980.Google Scholar
  17. [17]
    R.J.Lipton, D.J.Rose and R.E.Tarjan,“Generalized nested dissection”, SIAM J. Numer. Analysis 16(2),pp.346–358(1979).Google Scholar
  18. [18]
    V.Pan and J.Reif,“Parallel nested dissection for path algebra computations”, Operation research letters 5(4),pp.177–184,1986.Google Scholar
  19. [19]
    D.J.Rose, “A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equation”,Graph Theory and Computing,R.Read,ed., Academic Press,pp.183–217, 1972.Google Scholar
  20. [20]
    N.Robertson and P.D.Seymour,“Graph Minors III, algorithmic aspects of tree-width”,J. of Algorithms 7,1986,pp.309–322.Google Scholar
  21. [21]
    M.Schwartz,“Computer-communication network design and analysis”,Prentice Hall, 1977.Google Scholar
  22. [22]
    R.E.Stearns and H.B.Hunt III, “Power indices, structure trees, and easier hard problems”, SUNY Albany Tech. Rep. TR 89-21, 1989.Google Scholar
  23. [23]
    R.E.Tarjan, “A unified approach to path problems”, J. ACM 28(3),pp.594–614(1981)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Venkatesh Radhakrishnan
    • 1
  • Harry B. HuntIII
    • 1
  • Richard E. Stearns
    • 1
  1. 1.University at Albany, SunyUSA

Personalised recommendations