We present a simple shortest path algorithm. If the input lengths are positive and uniformly distributed, the algorithm runs in linear time. The worst-case running time of the algorithm is O(m + n log C), where n and m are the number of vertices and arcs of the input graph, respectively, and C is the ratio of the largest and the smallest nonzero arc length.

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 O(n + m) time RAM algorithm for undirected graphs with n nodes, m edges and integer edge weights in f0; : : : ; 2 w 1g where w denotes the word length, the currently best time bound for directed sparse graphs on a RAM is O(n + m log log n). 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 O(n + m) with high probability. 1 Introduction The single-source shortest-path problem (SSSP) is a fundamental and well-studied combinatorial optimization problem with many practical and theoretical applications [1]. Let G = (V; E) be a directed graph, jV j = n, jEj = m, let s be a distinguished vertex of the graph, and c be a function assigning a n...

We review shortest path algorithms based on the multi-level bucket data structure [6] and discuss the interplay between theory and engineering choices that leads to e#cient implementations. Our experimental results suggest that the caliber heuristic [17] and adaptive parameter selection give an e#cient algorithm, both on typical and on hard inputs, for a wide range of arc lengths.

We study the average-case complexity of shortest-paths problems in the vertex-potential model. The vertex-potential model is a family of probability distributions on complete directed graphs with arbitrary real edge lengths but without negative cycles. We show that on a graph with n vertices and with respect to this model, the single-source shortest-paths problem can be solved in O(n²) expected time, and the all-pairs shortest-paths problem can be solved in O(n² log n) expected time.

We present Filter-Kruskal – a simple modification of Kruskal’s algorithm that avoids sorting edges that are “obviously ” not in the MST. For arbitrary graphs with random edge weights Filter-Kruskal runs in time O ( m + n lognlog m n, i.e. in linear time for not too sparse graphs. Experiments indicate that the algorithm has very good practical performance over the entire range of edge densities. An equally simple parallelization seems to be the currently best practical algorithm on multicore machines. 1

We study both upper and lower bounds on the average-case complexity of shortestpaths algorithms. It is proved that the all-pairs shortest-paths problem on n-vertex networks can be solved in time O(n 2 log n) with high probability with respect to various probability distributions on the set of inputs. Our results include the first theoretical analysis of the average behavior of shortest-paths algorithms with respect to the vertex-potential model, a family of probability distributions on complete networks with arbitrary real arc costs but without negative cycles. We also generalize earlier work with respect to the common uniform model, and we correct the analysis of an algorithm with respect to the endpoint-independent model. For the algorithm that solves the all-pairs shortest-paths problem on networks generated according to the vertex-potential model, a key ingredient is an algorithm that solves the single-source shortest-paths problem on such networks in time O(n 2 ) with high probability. All algorithms mentioned exploit that with high probability, the single-source shortest-paths problem can be solved correctly by considering only a rather sparse subset of the arc set. We prove a lower bound indicating the limitations of this approach. In a fairly general probabilistic model, any algorithm solving the single-source shortest-paths problem has to inspect# n log n) arcs with high probability. Kurzzusammenfassung. In dieser Arbeit werden sowohl obere als auch untere Schranken f ur die average-case-Komplexit at von K urzeste-Wege-Algorithmen untersucht. Wir beweisen f ur verschiedene Wahrscheinlichkeitsverteilungen auf Netzwerken mit n Knoten, dass das all-pairs shortestpaths problem mit hoher Wahrscheinlichkeit in Zeit O(n 2 log n) gel ost werden kann. Insbeso...

Dijkstra's algorithm computes the shortest paths between a starting vertex and each other vertex in a directed graph. The performance of Dijkstra's algorithm depends on how it is implemented. This mainly relates to the type of data structure used for the frontier set.

We present an improvement of the multi-level bucket shortest path algorithm of Denardo and Fox [9] and justify this improvement, both theoretically and experimentally. We prove that if the input arc lengths come from a natural probability distribution, the new algorithm runs in linear average time while the original algorithm does not. We also describe an implementation of the new algorithm. Our experimental data suggests that the new algorithm is preferable to the original one in practice. Furthermore, for integral arc lengths that fit into a word of today's computers, the performance is close to that of breadth-first search, suggesting limitations on further practical improvements.

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 O(n + m) time RAM algorithm for undirected graphs with n nodes, m edges and integer edge weights in f0; : : : ; 2 1g where w denotes the word length, the currently best time bound for directed sparse graphs on a RAM is O(n +m log log n).