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Concurrent Threads and Optimal Parallel Minimum Spanning Trees Algorithm
 J. ACM
, 2001
"... This paper resolves a longstanding open problem on whether the concurrent write capability of parallel random access machine (PRAM) is essential for solving fundamental graph problems like connected components and minimum spanning trees in O(log n) time. Specically, we present a new algorithm to so ..."
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Cited by 25 (2 self)
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This paper resolves a longstanding open problem on whether the concurrent write capability of parallel random access machine (PRAM) is essential for solving fundamental graph problems like connected components and minimum spanning trees in O(log n) time. Specically, we present a new algorithm to solve these problems in O(log n) time using a linear number of processors on the exclusiveread exclusivewrite PRAM. The logarithmic time bound is actually optimal since it is well known that even computing the \OR" of n bits
Practical Parallel Algorithms for Minimum Spanning Trees
 In Workshop on Advances in Parallel and Distributed Systems
, 1998
"... We study parallel algorithms for computing the minimum spanning tree of a weighted undirected graph G with n vertices and m edges. We consider an input graph G with m=n p, where p is the number of processors. For this case, we show that simple algorithms with dataindependent communication patterns ..."
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Cited by 21 (0 self)
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We study parallel algorithms for computing the minimum spanning tree of a weighted undirected graph G with n vertices and m edges. We consider an input graph G with m=n p, where p is the number of processors. For this case, we show that simple algorithms with dataindependent communication patterns are efficient, both in theory and in practice. The algorithms are evaluated theoretically using Valiant's BSP model of parallel computation and empirically through implementation results.
CommunicationOptimal Parallel Minimum Spanning Tree Algorithms
, 1998
"... Lower and upper bounds for finding a minimum spanning tree (MST) in a weighted undirected graph on the BSP model are presented. We provide the first nontrivial lower bounds on the communication volume required to solve the MST problem. Let p denote the number of processors, n the number of nodes of ..."
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Cited by 13 (1 self)
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Lower and upper bounds for finding a minimum spanning tree (MST) in a weighted undirected graph on the BSP model are presented. We provide the first nontrivial lower bounds on the communication volume required to solve the MST problem. Let p denote the number of processors, n the number of nodes of the input graph, and m the number of edges of the input graph. We show that in the worst case, a total of \Omega\Gamma \Delta min(m; pn)) bits need to be communicated in order to solve the MST problem, where is the number of bits required to represent a single edge weight. This implies that if each message communicates at most bits, any BSP algorithm for finding an MST requires communication time \Omega\Gamma g \Delta min(m=p; n)), where g is the gap parameter of the BSP model. In addition, we present two algorithms with communication requirements that match our lower bound in different situations. Both algorithms perform linear work for appropriate values of n, m and p, and use a numbe...
A Randomized Linear Work EREW PRAM Algorithm to Find a Minimum Spanning Forest
, 1997
"... We present a randomized EREW PRAM algorithm to find a minimum spanning forest in a weighted undirected graph. On an nvertex graph the algorithm runs in o((log n) 1+ffl ) expected time for any ffl ? 0 and performs linear expected work. This is the first linear work, polylog time algorithm on th ..."
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Cited by 13 (2 self)
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We present a randomized EREW PRAM algorithm to find a minimum spanning forest in a weighted undirected graph. On an nvertex graph the algorithm runs in o((log n) 1+ffl ) expected time for any ffl ? 0 and performs linear expected work. This is the first linear work, polylog time algorithm on the EREW PRAM for this problem. This also gives parallel algorithms that perform expected linear work on two more realistic models of parallel computation, the QSM and the BSP. 1 Introduction The design of efficient algorithms to find a minimum spanning forest (MSF) in a weighted undirected graph is a fundamental problem that has received much attention. There have been many algorithms designed for the MSF problem that run in close to linear time (see, e.g., [CLR91]). Recently a randomized lineartime algorithm for this problem was presented in [KKT95]. Based on this work [CKT94] presented a randomized parallel algorithm on the CRCW PRAM which runs in O(2 log n log n) expected time whil...
Improved parallel algorithms for finding the most vital edge of a graph with respect to minimum spanning tree
, 1995
"... Let G be a connected, undirected and weighted graph with n vertices and m edges. A most vital edge of G with respect to minimum spanning tree is an edge whose removal from G will cause the greatest weightincrease in the minimum spanning tree of the remaining graph. This paper presents fast parallel ..."
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Let G be a connected, undirected and weighted graph with n vertices and m edges. A most vital edge of G with respect to minimum spanning tree is an edge whose removal from G will cause the greatest weightincrease in the minimum spanning tree of the remaining graph. This paper presents fast parallel algorithms that compute the most vital edge of G in O(log n) time using O(m log log log n = log n+n) CRCW processors, and in O(log n log log n) time using O((m + n 2 = log n) = log log n) CREW processors, respectively. This improves the known results of O(log n) time and O(m) processors on CRCW PRAM [11, 13], and of O(n) time and O(n 2 = log 2 n) processors on CREW PRAM [13]. Keywords: Minimum spanning tree, most vital edge, parallel algorithm, PRAM. 1
An Optimal Parallel Algorithm for Minimum Spanning Trees in Planar Graphs
"... Abstract. We present an optimal deterministic O(n)work parallel algorithm for finding a minimum spanning tree on an nvertex planar graph. The algorithm runs in O(log n) time on a CRCW PRAM and in O(log n log ∗ n) time on an EREW PRAM. Our results hold for any sparse graph that is closed under tak ..."
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Abstract. We present an optimal deterministic O(n)work parallel algorithm for finding a minimum spanning tree on an nvertex planar graph. The algorithm runs in O(log n) time on a CRCW PRAM and in O(log n log ∗ n) time on an EREW PRAM. Our results hold for any sparse graph that is closed under taking of minors, as well as for a class of graphs with nonbounded genus. 1