We present a deterministic algorithm for the connectivity problem on undirected graphs that runs in O(log 1:5 n) space. Thus, the recursive doubling technique of Savich which requires O(log^1.5 n) space is not optimal for this problem.

We present a simple and implementable algorithm that computes a minimum spanning tree of an undirected weighted graph G = (V, E) of n = |V| vertices and m = |E| edges on an EREW PRAM in O(log 3=2 n) time using n+m processors. This represents a substantial improvement in the running time over the previous results for this problem using at the same time the weakest of the PRAM models. It also implies the existence of algorithms having the same complexity bounds for the EREW PRAM, for connectivity, ear decomposition, biconnectivity, strong orientation, st-numbering and Euler tours problems.

Finding the connected components of an undirected graph G = (V; E) on n = jV j vertices and m = jEj edges is a fundamental computational problem. The best known parallel algorithm for the CREW PRAM model runs in O(log 2 n) time using n 2 = log 2 n processors [6, 15]. For the CRCW PRAM model, in which concurrent writing is permitted, the best known algorithm runs in O(log n) time using slightly more than (n +m)= log n processors [26, 9, 5]. Simulating this algorithm on the weaker CREW model increases its running time to O(log 2 n) [10, 19, 29]. We present here a simple algorithm that runs in O(log 3=2 n) time using n +m CREW processors. Finding an o(log 2 n) parallel connectivity algorithm for this model was an open problem for many years. 1 Introduction Let G = (V; E) be an undirected graph on n = jV j vertices and m = jEj edges. A path p of length k is a sequence of edges (e 1 ; \Delta \Delta \Delta ; e i ; \Delta \Delta \Delta ; e k ) such that e i 2 E for i = 1; \...

Improving a long chain of works we obtain a randomised EREW PRAM algorithm for finding the connected components of a graph G = (V; E) with n vertices and m edges in O(logn) time using an optimal number of O((m + n)= log n) processors. The result returned by the algorithm is always correct. The probability that the algorithm will not complete in O(log n) time is o(n \Gammac ) for any c ? 0. 1 Introduction Finding the connected components of an undirected graph is perhaps the most basic algorithmic graph problem. While the problem is trivial in the sequential setting, it seems that elaborate methods should be used to solve the problem efficiently in the parallel setting. A considerable number of researchers investigated the complexity of the problem in various parallel models including, in particular, various members of the PRAM family. In this work we consider the EREW PRAM model, the weakest member of this family, and obtain, for the first time, a parallel connectivity algorith...

We present the first randomized O(log n) time and O(m+n) work EREW PRAM algorithm for finding a spanning forest of an undirected graph G = (V; E) with n vertices and m edges. Our algorithm is optimal with respect to time, work and space. As a consequence we get optimal randomized EREW PRAM algorithms for other basic connectivity problems such as finding a bipartite partition, finding bridges and biconnected components, finding Euler tours in Eulerian graphs, finding an ear decomposition, finding an open ear decomposition, finding a strong orientation, and finding an st-numbering.

We show that if a planar graph has a simple cycle of length k, where k is a fixed integer, such a cycle may be computed in O(log n) time by a randomized EREW-PRAM with O(n) processors with high probability. This improves a previous result of [8]. The improvement relies on an efficient parallel algorithm for computing a large independent set in a constant-degree-bounded conflict graph, which is a natural method to avoid memory access conflicts in EREW-PRAM graph algorithms. Many EREW-PRAM algorithms use results from [6], [11], which can be used to compute such a set in O(log n) parallel time. This paper gives an O(1) time randomized algorithm using O(n) processors for that problem. This method can also be used to improve the randomized running time of many other EREW-PRAM algorithms. 1 Introduction It is well known that finding the longest cycle in a graph is a hard problem, since finding a hamiltonian cycle is NP-complete [10]. Hence finding a simple cycle of lenght k, for an arbi...