Single-source shortest-paths on arbitrary directed graphs in linear average-case time

The quest for a linear-time single-source shortest-path (SSSP) algorithm on directed graphs with positive edge weights is an ongoing hot research topic. While Thorup recently found an <i>&Ogr;</i>(<i>n</i> + <i>m</i>) time RAM algorithm for undirected graphs with <i>n</i> nodes, <i>m</i> edges and integer edge weights in {0,…,2<sup>w</sup> - 1} where <i>w</i> denotes the word length, the currently best time bound for directed sparse graphs on a RAM is <i>&Ogr;</i>(<i>n</i> + <i>m</i> · log log <i>n</i>). In the present paper we study the average-case complexity of SSSP. We give a simple algorithm for arbitrary directed graphs with random edge weights uniformly distributed in [0, 1] and show that it needs linear time <i>&Ogr;</i>(<i>n</i> + <i>m</i>) with high probability.

[1]  Rajeev Raman,et al.  Recent results on the single-source shortest paths problem , 1997, SIGA.

[2]  K. Athreya,et al.  Large Deviation Rates for Branching Processes--I. Single Type Case , 1994 .

[3]  Mikkel Thorup,et al.  On RAM priority queues , 1996, SODA '96.

[4]  Philip M. Spira,et al.  A New Algorithm for Finding all Shortest Paths in a Graph of Positive Arcs in Average Time 0(n2 log2n) , 1973, SIAM J. Comput..

[5]  Kohei Noshita,et al.  A Theorem on the Expected Complexity of Dijkstra's Shortest Path Algorithm , 1985, J. Algorithms.

[6]  Andrew V. Goldberg,et al.  Shortest paths algorithms: Theory and experimental evaluation , 1994, SODA '94.

[7]  Robert B. Dial,et al.  Algorithm 360: shortest-path forest with topological ordering [H] , 1969, CACM.

[8]  M. Habib.  Probabilistic methods for algorithmic discrete mathematics , 1998 .

[9]  C. McDiarmid.  Concentration , 1862, The Dental register.

[10]  Refael Hassin,et al.  On Shortest Paths in Graphs with Random Weights , 1985, Math. Oper. Res..

[11]  E. Denardo,et al.  Shortest-Route Methods: 1. Reaching, Pruning, and Buckets , 1979, Oper. Res..

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

[13]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[14]  Mikkel Thorup,et al.  Undirected single-source shortest paths with positive integer weights in linear time , 1999, JACM.

[15]  Kurt Mehlhorn,et al.  On the all-pairs shortest-path algorithm of Moffat and Takaoka , 1997, Random Struct. Algorithms.

[16]  Stefano Pallottino,et al.  Shortest-path methods: Complexity, interrelations and new propositions , 1984, Networks.

[17]  Rajeev Raman,et al.  Priority Queues: Small, Monotone and Trans-dichotomous , 1996, ESA.

[18]  F. Benjamin Zhan,et al.  Shortest Path Algorithms: An Evaluation Using Real Road Networks , 1998, Transp. Sci..

[19]  Kurt Mehlhorn,et al.  On the all-pairs shortest-path algorithm of Moffat and Takaoka , 1997 .

[20]  Alan M. Frieze,et al.  Average-Case Complexity of Shortest-Paths Problems in the Vertex-Potential Model , 1997, RANDOM.

[21]  Alan M. Frieze,et al.  The shortest-path problem for graphs with random arc-lengths , 1985, Discret. Appl. Math..

[22]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[23]  Alistair Moffat,et al.  An All Pairs Shortest Path Algorithm with Expected Time O(n² log n) , 1987, SIAM J. Comput..

[24]  Torben Hagerup,et al.  Improved Shortest Paths on the Word RAM , 2000, ICALP.