Shortest Paths in Digraphs of Small Treewidth. Part I: Sequential Algorithms

Abstract. We consider the problem of preprocessing an n -vertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give algorithms that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms can answer distance queries in O(α(n)) time after O(n) preprocessing. This improves upon previously known results for the same problem. We also give a dynamic algorithm which, after a change in an edge weight, updates the data structure in time O(nβ) , for any constant 0 < β < 1 . Furthermore, an algorithm of independent interest is given: computing a shortest path tree, or finding a negative cycle in linear time.

[1]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[2]  Alberto Marchetti-Spaccamela,et al.  Dynamic algorithms for shortest paths in planar graphs , 1993, Theor. Comput. Sci..

[3]  Hans L. Bodlaender,et al.  A Tourist Guide through Treewidth , 1993, Acta Cybern..

[4]  Paul D. Seymour,et al.  Graph Minors. II. Algorithmic Aspects of Tree-Width , 1986, J. Algorithms.

[5]  Grammati E. Pantziou,et al.  On-line and Dynamic Algorithms for Shorted Path Problems , 1995, STACS.

[6]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

[7]  Paul D. Seymour,et al.  Graph Minors: XV. Giant Steps , 1996, J. Comb. Theory, Ser. B.

[8]  Greg N. Frederickson,et al.  Fast Algorithms for Shortest Paths in Planar Graphs, with Applications , 1987, SIAM J. Comput..

[9]  Bernard Chazelle.  Computing on a Free Tree via Complexity-Preserving Mappings , 1984, FOCS.

[10]  Hans L. Bodlaender.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC '93.

[11]  Paul G. Spirakis,et al.  Efficient Sequential and Parallel Algorithms for the Negative Cycle Problem , 1994, ISAAC.

[12]  Stefan Arnborg,et al.  Efficient algorithms for combinatorial problems on graphs with bounded decomposability — A survey , 1985, BIT.

[13]  Greg N. Frederickson,et al.  Planar graph decomposition and all pairs shortest paths , 1991, JACM.

[14]  Greg N. Frederickson.  Using Cellular Graph Embeddings in Solving All Pairs Shortest Paths Problems , 1995, J. Algorithms.

[15]  W. Ackermann.  Zum Hilbertschen Aufbau der reellen Zahlen , 1928 .

[16]  Philip N. Klein,et al.  Faster Shortest-Path Algorithms for Planar Graphs , 1997, J. Comput. Syst. Sci..

[17]  Detlef Seese,et al.  Easy Problems for Tree-Decomposable Graphs , 1991, J. Algorithms.

[18]  Hans L. Bodlaender,et al.  Dynamic Algorithms for Graphs with Treewidth 2 , 1993, WG.