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 show how to decompose efficiently in parallel any graph into a number, ~ fl, of outerplanar subgraphs (called hammocks) satisfying certain separator properties. Our work combines and extends the sequential hammock decomposition technique introduced by G. Frederickson and the parallel ear decomposition technique, thus we call it the hammock-on-ears decomposition. We mention that hammock-on-ears decomposition also draws from techniques in computational geometry and that an embedding of the graph does not need to be provided with the input. We achieve this decomposition in O(logn log log n) time using O(n + m) CREW PRAM processors, for an n-vertex, m-edge graph or digraph. The hammock-on-ears decomposition implies a general framework for solving graph problems efficiently. Its value is demonstrated by a variety of applications on a significant class of (di)graphs, namely that of sparse (di)graphs. This class consists of all (di)graphs which have a ~ fl between 1 and \Theta(n...

The girth of a graph G has been de ned as the length of a shortest cycle of G. We design an O(n 5=4 log n) algorithm for finding the girth of an undirected n-vertex planar graph, giving the first o(n 2 ) algorithm for this problem. Our approach combines several techniques such as graph separation, hammock decomposition, covering of a planar graph with graphs of small tree-width, and dynamic shortest path computation. We discuss extensions and generalizations of our result.

Our work is based on the pioneering work in sphere separators done by Miller, Teng, Vavasis et al, [8, 12], who gave efficient static (fixed input) algorithms for finding sphere separators of size s(n) = O(n d ) for a set of points in R . We present

We examine the problem of transmitting in minimum time a given amount of data between a source and a destination in a network with finite channel capacities and non--zero propagation delays. In the absence of delays, the problem has been shown to be solvable in polynomial time. In this paper, we show that the general problem is NP--complete. In addition, we examine transmissions along a single path, called the quickest path, and present algorithms for general and special classes of networks that improve upon previous approaches. The first dynamic algorithm for the quickest path problem is also given. Keywords: Data transmission, sparse network, transmission time, quickest path, dynamic algorithm. 1 Introduction Consider an n-node, m-edge network N = (V; E; c; l), where G = (V; E) is a directed graph, c : E ! IR + is the capacity function and l : E ! IR is the delay function. The nodes represent transmitters/receivers without data memories and the edges represent communication ...

We generalize all the results obtained for maximum integer multiflow and minimum multicut problems in trees by Garg et al. [Primaldual approximation algorithms for integral flow and multicut in trees. Algorithmica 18 (1997) 3–20] to graphs with a fixed cyclomatic number, while this cannot be achieved for other classical generalizations of the trees. Moreover, we prove that the minimum multicut problem with a fixed number of source-sink pairs is polynomial-time solvable in planar and in bounded tree-width graphs. Eventually, we introduce the class of k-edge-outerplanar graphs and show that the integrality gap of the maximum edge-disjoint paths problem is bounded in these graphs. We also provide stronger results for cacti (k = 1).

This paper presents experiences from the implementation of parallel algorithms for the all-pairs shortest path problem. A brief survey of the problem is given and efficient techniques for exploiting the memory hierarchy of parallel computers are described. Comparing the algorithms at a practical level, the importance of issues such as synchronisation and data locality is established. 1 Introduction The problem of finding the shortest path in a graph is one of the most important problems in operations research. It arises in many applications and, as a result, has been the subject of numerous studies which provide solutions for both sequential [7] and parallel [3] computers. This paper deals with algorithms for solving the allpairs shortest path problem (i.e. finding the shortest paths between all pairs of nodes in a graph) on parallel computers. While theoretical measures of the performance of these algorithms are rather well described, it is well known that for parallel implement...

We present linear-I/O algorithms for fundamental graph problems on embedded outerplanar graphs. We show that breadth-first search, depth-first search, single-source shortest paths, triangulation, and computing an ɛseparator of size O(1/ɛ) takeO(scan(N)) I/Os on embedded outerplanar graphs. We also show that it takes O(sort(N)) I/Os to test whether a given graph is outerplanar and to compute an outerplanar embedding of an outerplanar graph, thereby providing O(sort(N))-I/O algorithms for the above problems if no embedding of the graph is given. As all these problems have linear-time algorithms in internal memory, a simple simulation technique can be used to improve the I/O-complexity of our algorithms from O(sort(N)) to O(perm(N)). We prove matching lower bounds for embedding, breadth-first search, depth-first search, and singlesource shortest paths if no embedding is given. Our algorithms for the above problems use a simple linear-I/O time-forward processing algorithm for rooted trees whose vertices are stored in preorder.

We consider the well known geometric problem of determining shortest paths between pairs of points on a polyhedral surface P, where P consists of triangular faces with positive weights assigned to them. The cost of a path in P is defined to be the weighted sum of Euclidean lengths of the sub-paths within each face of P. We present query algorithms that compute approximate distances and/or approximate shortest paths on P. Our all-pairs query algorithms take as input an approximation parameter ε ∈ (0,1) and a query time parameter q, in a certain range, and builds a data structure APQ(P,ε;q), which is then used for answering ε-approximate distance queries in O(q) time. As a building block of the APQ(P,ε;q) data structure, we develop a single source query data structure SSQ(a;P,ε) that can answer ε-approximate distance queries from a fixed point a to any query point on P in logarithmic time. Our algorithms answer shortest path queries in weighted surfaces, which is an important extension, both theoretically and practically, to the extensively studied Euclidean distance case. In addition, our algorithms improve upon previously known query algorithms for shortest paths on surfaces. The algorithms are based on a novel graph separator algorithm introduced and analyzed here, which extends and generalizes previously known separator algorithms.

Abstract. We give an O(n log n) algorithm for computing the girth (shortest cycle) of an undirected n-vertex planar graph. Our solution extends to any graph of bounded genus. This improves upon the best previously known algorithms for this problem. Key words. Girth, shortest cycle, planar graph, graphs of bounded genus AMS subject classifications. 05C38, 68R10