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A Randomized LinearTime Algorithm to Find Minimum Spanning Trees
, 1994
"... We present a randomized lineartime algorithm to find a minimum spanning tree in a connected graph with edge weights. The algorithm uses random sampling in combination with a recently discovered lineartime algorithm for verifying a minimum spanning tree. Our computational model is a unitcost ra ..."
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Cited by 115 (7 self)
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We present a randomized lineartime algorithm to find a minimum spanning tree in a connected graph with edge weights. The algorithm uses random sampling in combination with a recently discovered lineartime algorithm for verifying a minimum spanning tree. Our computational model is a unitcost randomaccess machine with the restriction that the only operations allowed on edge weights are binary comparisons.
On externalmemory MST, SSSP and multiway planar graph separation
 In Proc. 8th Scandinavian Workshop on Algorithmic Theory, volume 1851 of LNCS
, 2000
"... Recently external memory graph algorithms have received considerable attention because massive graphs arise naturally in many applications involving massive data sets. Even though a large number of I/Oefficient graph algorithms have been developed, a number of fundamental problems still remain open ..."
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Cited by 24 (2 self)
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Recently external memory graph algorithms have received considerable attention because massive graphs arise naturally in many applications involving massive data sets. Even though a large number of I/Oefficient graph algorithms have been developed, a number of fundamental problems still remain open. In this paper we develop an improved algorithm for the problem of computing a minimum spanning tree of a general graph, as well as new algorithms for the single source shortest paths and the multiway graph separation problems on planar graphs.
A linearwork parallel algorithm for finding . . .
, 1994
"... We give the first linearwork parallel algorithm for finding a minimum spanning tree. It is a randomized algorithm, and requires O(2log \Lambda n log n) expected time. It is a modification of the sequential lineartime algorithm of Klein and Tarjan. ..."
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Cited by 14 (1 self)
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We give the first linearwork parallel algorithm for finding a minimum spanning tree. It is a randomized algorithm, and requires O(2log \Lambda n log n) expected time. It is a modification of the sequential lineartime algorithm of Klein and Tarjan.
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 9 (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...
An Optimal EREW PRAM Algorithm For Minimum Spanning Tree Verification
, 1997
"... We present a deterministic parallel algorithm on the EREW PRAM model to verify a minimum spanning tree of a graph. The algorithm runs on a graph with n vertices and m edges in O(log n) time and O(m + n) work. The algorithm is a parallelization of King's linear time sequential algorithm for the prob ..."
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Cited by 7 (2 self)
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We present a deterministic parallel algorithm on the EREW PRAM model to verify a minimum spanning tree of a graph. The algorithm runs on a graph with n vertices and m edges in O(log n) time and O(m + n) work. The algorithm is a parallelization of King's linear time sequential algorithm for the problem. 1 Introduction The problem of verifying if a given spanning tree in an edgeweighted graph is a minimum spanning tree for the graph is an important problem which is closely related to the problem of finding a minimum spanning tree of a graph. Recently Dixon, Rauch and Tarjan ([DRT92]) and King ([Kin95]) have developed sequential linear time algorithms for the problem. Dixon and Tarjan have also given an optimal CREW algorithm ([DT94]). In this paper, we present a parallel algorithm which runs in optimal time and work bounds on the weaker EREW PRAM model. This resolves an open question posed in [DT94]. The highlevel structure of the algorithm has been adapted from [DT94] and [Kin95]. W...
On Minimum Spanning Trees
, 1998
"... The minimal spanning tree problem is one of the oldest and most basic graph problems in theoretical computer science. Its history dates back to Boruvka's algorithm in 1926 and till today it is a extensively researched problem with new breakthroughs only recently. Given an edgeweighted graph, this p ..."
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Cited by 3 (0 self)
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The minimal spanning tree problem is one of the oldest and most basic graph problems in theoretical computer science. Its history dates back to Boruvka's algorithm in 1926 and till today it is a extensively researched problem with new breakthroughs only recently. Given an edgeweighted graph, this problem calls for nding a subtree spanning all the vertices, whose total weight isminimal. The central open question is: Does there exist a linear time deterministic algorithm that finds the minimal spanning tree? This problem remains open. We will present the problem from different angles giving our insight on each. We present the problem in a new framework, and improve the running time for specific families of graphs in this framework.
Approximation Algorithms for Network Design: A Survey
"... In a typical instance of a network design problem, we are given a directed or undirected graph G = (V,E), nonnegative edgecosts ce for all e ∈ E, and our goal is to find a minimumcost subgraph H of G that satisfies some design criteria. For example, we may wish to find a minimumcost set of edges ..."
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Cited by 3 (0 self)
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In a typical instance of a network design problem, we are given a directed or undirected graph G = (V,E), nonnegative edgecosts ce for all e ∈ E, and our goal is to find a minimumcost subgraph H of G that satisfies some design criteria. For example, we may wish to find a minimumcost set of edges that induces a connected graph (this is the minimumcost spanning tree problem), or we might want to find a minimumcost set of arcs in a directed graph such that every vertex can reach every other vertex (this is the minimumcost strongly connected subgraph problem). This abstract model for network design problems has a large number of practical applications; the design process of telecommunication and traffic networks, and VLSI chip design are just two examples. Many practically relevant instances of network design problems are NPhard, and thus likely intractable. This survey focuses on approximation algorithms as one possible way of circumventing this impasse. Approximation algorithms are efficient (i.e., they run in polynomialtime), and they compute solutions to a given instance of an optimization problem whose objective values are close to those of the respective optimum solutions. More concretely, most of the problems discussed in this survey are minimization problems. We then say that an algorithm is an αapproximation for a given problem if the ratio of the cost of an approximate solution computed by the algorithm to that of an optimum solution is at most α over all instances. In the