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19
A Functional Approach to External Graph Algorithms
 Algorithmica
, 1998
"... . We present a new approach for designing external graph algorithms and use it to design simple external algorithms for computing connected components, minimum spanning trees, bottleneck minimum spanning trees, and maximal matchings in undirected graphs and multigraphs. Our I/O bounds compete w ..."
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Cited by 90 (2 self)
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. We present a new approach for designing external graph algorithms and use it to design simple external algorithms for computing connected components, minimum spanning trees, bottleneck minimum spanning trees, and maximal matchings in undirected graphs and multigraphs. Our I/O bounds compete with those of previous approaches. Unlike previous approaches, ours is purely functionalwithout side effectsand is thus amenable to standard checkpointing and programming language optimization techniques. This is an important practical consideration for applications that may take hours to run. 1 Introduction We present a divideandconquer approach for designing external graph algorithms, i.e., algorithms on graphs that are too large to fit in main memory. Our approach is simple to describe and implement: it builds a succession of graph transformations that reduce to sorting, selection, and a recursive bucketing technique. No sophisticated data structures are needed. We apply our t...
Parallel SymmetryBreaking in Sparse Graphs
 SIAM J. Disc. Math
, 1987
"... We describe efficient deterministic techniques for breaking symmetry in parallel. These techniques work well on rooted trees and graphs of constant degree or genus. Our primary technique allows us to 3color a rooted tree in O(lg n) time on an EREW PRAM using a linear number of processors. We use th ..."
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Cited by 73 (2 self)
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We describe efficient deterministic techniques for breaking symmetry in parallel. These techniques work well on rooted trees and graphs of constant degree or genus. Our primary technique allows us to 3color a rooted tree in O(lg n) time on an EREW PRAM using a linear number of processors. We use these techniques to construct fast linear processor algorithms for several problems, including (\Delta + 1)coloring constantdegree graphs and 5coloring planar graphs. We also prove lower bounds for 2coloring directed lists and for finding maximal independent sets in arbitrary graphs. 1 Introduction Some problems for which trivial sequential algorithms exist appear to be much harder to solve in a parallel framework. When converting a sequential algorithm to a parallel one, at each step of the parallel algorithm we have to choose a set of operations which may be executed in parallel. Often, we have to choose these operations from a large set A preliminary version of this paper appear...
Removing Randomness in Parallel Computation Without a Processor Penalty
 Journal of Computer and System Sciences
, 1988
"... We develop some general techniques for converting randomized parallel algorithms into deterministic parallel algorithms without a blowup in the number of processors. One of the requirements for the application of these techniques is that the analysis of the randomized algorithm uses only pairwise in ..."
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Cited by 49 (1 self)
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We develop some general techniques for converting randomized parallel algorithms into deterministic parallel algorithms without a blowup in the number of processors. One of the requirements for the application of these techniques is that the analysis of the randomized algorithm uses only pairwise independence. Our main new result is a parallel algorithm for coloring the vertices of an undirected graph using at most \Delta + 1 distinct colors in such a way that no two adjacent vertices receive the same color, where \Delta is the maximum degree of any vertex in the graph. The running time of the algorithm is O(log 3 n log log n) using a linear number of processors on a concurrent read, exclusive write (CREW) parallel random access machine (PRAM). 1 Our techniques also apply to several other problems, including the maximal independent set problem and the maximal matching problem. The application of the general technique to these last two problems is mostly of academic interest because...
An NC Algorithm for Minimum Cuts
 IN PROCEEDINGS OF THE 25TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
"... We show that the minimum cut problem for weighted undirected graphs can be solved in NC using three separate and independently interesting results. The first is an (m 2 =n)processor NC algorithm for finding a (2 + ffl)approximation to the minimum cut. The second is a randomized reduction from ..."
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Cited by 46 (3 self)
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We show that the minimum cut problem for weighted undirected graphs can be solved in NC using three separate and independently interesting results. The first is an (m 2 =n)processor NC algorithm for finding a (2 + ffl)approximation to the minimum cut. The second is a randomized reduction from the minimum cut problem to the problem of obtaining a (2 + ffl)approximation to the minimum cut. This reduction involves a natural combinatorial SetIsolation Problem that can be solved easily in RNC. The third result is a derandomization of this RNC solution that requires a combination of two widely used tools: pairwise independence and random walks on expanders. We believe that the setisolation approach will prove useful in other derandomization problems. The techniques extend to two related problems: we describe NC algorithms finding minimum kway cuts for any constant k and finding all cuts of value within any constant factor of the minimum. Another application of these techni...
A New Parallel Algorithm For The Maximal Independent Set Problem
, 1989
"... A new parallel algorithm for the maximal independent set problem is constructed. It runs in O(log 4 n) time when implemented on a linear number of EREWprocessors. This is the first deterministic algorithm for the maximal independent set problem (MIS) whose running time is polylogarithmic and whose ..."
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Cited by 35 (2 self)
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A new parallel algorithm for the maximal independent set problem is constructed. It runs in O(log 4 n) time when implemented on a linear number of EREWprocessors. This is the first deterministic algorithm for the maximal independent set problem (MIS) whose running time is polylogarithmic and whose processortime product is optimal up to a polylogarithmic factor.
Parallel Algorithmic Techniques for Combinatorial Computation
 Ann. Rev. Comput. Sci
, 1988
"... this paper and supplied many helpful comments. This research was supported in part by NSF grants DCR8511713, CCR8605353, and CCR8814977, and by DARPA contract N0003984C0165. ..."
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Cited by 29 (3 self)
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this paper and supplied many helpful comments. This research was supported in part by NSF grants DCR8511713, CCR8605353, and CCR8814977, and by DARPA contract N0003984C0165.
Numerical Computation Of A Polynomial GCD And Extensions
, 1996
"... In the first part of this paper, we dene approximate polynomial gcds (greatest common divisors) and extended gcds provided that approximations to the zeros of the input polynomials are available. We relate our novel definition to the older and weaker ones, based on perturbation of the coefficients o ..."
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Cited by 24 (8 self)
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In the first part of this paper, we dene approximate polynomial gcds (greatest common divisors) and extended gcds provided that approximations to the zeros of the input polynomials are available. We relate our novel definition to the older and weaker ones, based on perturbation of the coefficients of the input polynomials, we demonstrate some deficiency of the latter definitions (which our denition avoids), and we propose new effective sequential and parallel (RNC and NC) algorithms for computing approximate gcds and extended gcds. Our stronger results are obtained with no increase of the asymptotic bounds on the computational cost. This is partly due to application of our recent nearly optimal algorithms for approximating polynomial zeros. In the second part of our paper, working under the older and more customary definition of approximate gcds, we modify and develop an alternative approach, which was previously based on the computation of the Singular Value Decomposition (SVD) of the associat...
Parallel Approximation Algorithms for Maximum Weighted Matching in General Graphs
 Information Processing Letters
, 2000
"... . The problem of computing a matching of maximum weight in a given edgeweighted graph is not known to be Phard or in RNC. This paper presents four parallel approximation algorithms for this problem. The first is an RNCapproximation scheme, i.e., an RNC algorithm that computes a matching of weight ..."
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Cited by 15 (0 self)
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. The problem of computing a matching of maximum weight in a given edgeweighted graph is not known to be Phard or in RNC. This paper presents four parallel approximation algorithms for this problem. The first is an RNCapproximation scheme, i.e., an RNC algorithm that computes a matching of weight at least 10 ffl times the maximum for any given constant ffl ? 0. The second one is an NC approximation algorithm achieving an approximation ratio of 1 2+ffl for any fixed ffl ? 0. The third and fourth algorithms only need to know the total order of weights, so they are useful when the edge weights require a large amount of memories to represent. The third one is an NC approximation algorithm that finds a matching of weight at least 2 31+2 times the maximum, where 1 is the maximum degree of the graph. The fourth one is an RNC algorithm that finds a matching of weight at least 1 21+4 times the maximum on average, and runs in O(log 1) time, not depending on the size of the graph. Key word...
On the parallel complexity of Hamiltonian Cycle and Matching Problem on Dense Graphs
 Journal of Algorithms
, 1993
"... Dirac's classical theorem asserts that, if every vertex of a graph G on n vertices has degree at least n 2 then G has a Hamiltonian cycle. We give a fast parallel algorithm on a CREW \Gamma PRAM to find a Hamiltonian cycle in such graphs. Our algorithm uses a linear number of processors and is o ..."
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Cited by 11 (1 self)
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Dirac's classical theorem asserts that, if every vertex of a graph G on n vertices has degree at least n 2 then G has a Hamiltonian cycle. We give a fast parallel algorithm on a CREW \Gamma PRAM to find a Hamiltonian cycle in such graphs. Our algorithm uses a linear number of processors and is optimal up to a polylogarithmic factor. The algorithm works in O(log 4 n) parallel time and uses linear number of processors on a CREW \Gamma PRAM . Our method bears some resemblance to Anderson's RNC algorithm [An] for maximal paths: we, too, start from a system of disjoint paths and try to glue them together. We are, however, able to perform the base step (perfect matching) deterministically. We also prove that a perfect matching in dense graphs can be found in NC 2 . The cost of improved time is a quadratic number of processors. On the negative side, we prove that finding an NC algorithm for perfect matching in slightly less dense graphs (minimum degree is at least ( 1 2 \Gamma ff...
Parallel Algorithms for Finding Maximal KDependent Sets and Maximal FMatchings
, 1991
"... Let k be a positive integer, a subset Q of the set of vertices of a graph G is k dependent in G if each vertex of Q has no more than k neighbours in Q. We present a parallel algorithm which computes a maximal kdependent set in a graph on n nodes in time O(log 4 n) on an EREW PRAM with O(n 2 ..."
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Cited by 9 (3 self)
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Let k be a positive integer, a subset Q of the set of vertices of a graph G is k dependent in G if each vertex of Q has no more than k neighbours in Q. We present a parallel algorithm which computes a maximal kdependent set in a graph on n nodes in time O(log 4 n) on an EREW PRAM with O(n 2 ) processors. In this way, we establish the membership of the problem of constructing a maximal kdependent set in the class NC. Our algorithm can be easily adapted to compute a maximal kdependent set in a graph of bounded valence in time O(log n) using only O(n) EREW PRAM processors. Let f be a positive integer function defined on the set V of vertices of a graph G: A subset F of the set of edges of G is said to be an fmatching if every vertex v 2 V is adjacent to at most f(v) edges in F . We present the first NC algorithm for constructing a maximal fmatching. For a graph on n nodes and m edges the algorithm runs in time O(log 4 n) and uses O(n+m) EREW PRAM processors. For gr...