Let G = (V; E) be an undirected weighted graph with jV j = n and jEj = m. Let k 1 be an integer. We show that G = (V; E) can be preprocessed in O(kmn ) expected time, constructing a data structure of size O(kn ), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k \Gamma 1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k \Gamma 1. We show that a 1963 girth conjecture of Erdos, implies ) space is needed in the worst case for any real stretch strictly smaller than 2k + 1. The space requirement of our algorithm is, therefore, essentially optimal.

We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms. The first algorithm solves...

The single source shortest paths problem (SSSP) is one of the classic problems in algorithmic graph theory: given a weighted graph G with a source vertex s, find the shortest path from s to all other vertices in the graph. Since 1959 all theoretical developments in SSSP have been based on Dijkstra's algorithm, visiting the vertices in order of increasing distance from s. Thus, any implementation of Dijkstra 's algorithm sorts the vertices according to their distances from s. However, we do not know how to sort in linear time. Here, a deterministic linear time and linear space algorithm is presented for the undirected single source shortest paths problem with integer weights. The algorithm avoids the sorting bottle-neck by building a hierechical bucketing structure, identifying vertex pairs that may be visited in any order. 1 Introduction Let G = (V; E), jV j = n, jEj = m, be an undirected connected graph with an integer edge weight function ` : E ! N and a distinguished source vertex...

We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems.

In this paper we consider the problem of computing a minimum cycle basis in a graph G with m edges and n vertices. The edges of G have non-negative weights on them. The previous best result for this problem was an O(m ω n) algorithm, where ω is the best exponent of matrix multiplication. It is presently known that ω < 2.376. We obtain an O(m 2 n + mn 2 log n) algorithm for this problem. Our algorithm also uses fast matrix multiplication. When the edge weights are integers, we have an O(m 2 n) algorithm. For unweighted graphs which are reasonably dense, our algorithm runs in O(m ω) time. For any ɛ> 0, we also design a 1 + ɛ approximation algorithm to compute a cycle basis which is at most 1 + ɛ times the weight of a minimum cycle basis. The running time of this algorithm is O ( mω ɛ log(W/ɛ)) for reasonably dense graphs, where W is the largest edge weight.

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Abstract. We show how to compute single-source shortest paths in undirected graphs with non-negative edge lengths in O ( p nm/B log n + MST (n, m)) I/Os, where n is the number of vertices, m is the number of edges, B is the disk block size, and MST (n, m) is the I/O-cost of computing a minimum spanning tree. For sparse graphs, the new algorithm performs O((n / √ B) log n) I/Os. This result removes our previous algorithm’s dependence on the edge lengths in the graph. 1

Abstract. We present a new scheme for computing shortest paths on real-weighted undirected graphs in the fundamental comparison-addition model. In an efficient preprocessing phase our algorithm creates a linear-size structure that facilitates single-source shortest path computations in O(m log α) time, where α = α(m, n) is the very slowly growing inverse-Ackermann function, m the number of edges, and n the number of vertices. As special cases our algorithm implies new bounds on both the all-pairs and single-source shortest paths problems. We solve the all-pairs problem in O(mnlog α(m, n)) time and, if the ratio between the maximum and minimum edge lengths is bounded by n (log n)O(1) , we can solve the single-source problem in O(m + nlog log n) time. Both these results are theoretical improvements over Dijkstra’s algorithm, which was the previous best for real weighted undirected graphs. Our algorithm takes the hierarchy-based approach invented by Thorup. Key words. single-source shortest paths, all-pairs shortest paths, undirected graphs, Dijkstra’s

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...

We consider the problem of computing a minimum cycle basis of an undirected edge-weighted graph G with m edges and n vertices. In this problem, a {0, 1} incidence vector is associated with each cycle and the vector space over F2 generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis of its cycle space. A cycle basis where the sum of the weights of its cycles is minimum is called a minimum cycle basis of G. Minimum cycle bases are useful in a number of contexts, e.g., the analysis of electrical networks, structural engineering, and chemistry. We present an O(m 2 n + mn 2 log n) algorithm to compute a minimum cycle basis. The previously best known running time to compute a minimum cycle basis was O(m ω n), where ω is the exponent of matrix multiplication. It is presently known that ω < 2.376. When the edge weights are integers, we give an O(m 2 n) algorithm. For unweighted graphs which are reasonably