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A simple parallel algorithm for the maximal independent set problem. (1986)

by M Luby
Venue:SIAM Journal on Computing,
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Parallel Multilevel k-way Partitioning Scheme for Irregular Graphs

by George Karypis, Vipin Kumar , 1996
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Abstract - Cited by 634 (32 self) - Add to MetaCart
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...tinct colors used is small. Our parallel graph coloring algorithm consists of a number of iterations. In each iteration a maximal independent set of vertices I is selected using a variation of Luby’s =-=[27]-=- algorithm. All vertices in this independent set are assigned the same color. Before the next iteration begins, the vertices in I are removed from the graph, and this smaller graph becomes the input g...

Simple Constructions of Almost k-wise Independent Random Variables

by Noga Alon, Oded Goldreich, Johan Håstad, René Peralta , 1992
"... We present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is (2 + o(1))(log log n + k/2 + log k + log 1 ɛ), where ɛ is the statistical difference between the dist ..."
Abstract - Cited by 303 (40 self) - Add to MetaCart
We present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is (2 + o(1))(log log n + k/2 + log k + log 1 ɛ), where ɛ is the statistical difference between the distribution induced on any k bit locations and the uniform distribution. This is asymptotically comparable to the construction recently presented by Naor and Naor (our size bound is better as long as ɛ < 1/(k log n)). An additional advantage of our constructions is their simplicity.

Small-Bias Probability Spaces: Efficient Constructions and Applications

by Joseph Naor, Moni Naor - SIAM J. Comput , 1993
"... We show how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with "almost" equal probability. They are called ffl-biased random variables. The number of random bits needed to generate the random var ..."
Abstract - Cited by 276 (13 self) - Add to MetaCart
We show how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with "almost" equal probability. They are called ffl-biased random variables. The number of random bits needed to generate the random variables is O(log n + log 1 ffl ). Thus, if ffl is polynomially small, then the size of the sample space is also polynomial. Random variables that are ffl-biased can be used to construct "almost" k-wise independent random variables where ffl is a function of k. These probability spaces have various applications: 1. Derandomization of algorithms: many randomized algorithms that require only k- wise independence of their random bits (where k is bounded by O(log n)), can be derandomized by using ffl-biased random variables. 2. Reducing the number of random bits required by certain randomized algorithms, e.g., verification of matrix multiplication. 3. Exhaustive testing of combinatorial circui...
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...o the problem of reducing the number of random bits in a randomized algorithm is to show that limited independence of the random variables suffices to assure a high probability of success. (See e.g., =-=[5, 36, 38]-=-). However, in our case, it is not clear from the proof of Proposition 3.1 how many vectors r remain distinguishers with respect to the vector v when the entries of R are not completely independent. M...

Programming Parallel Algorithms

by Guy E. Blelloch , 1996
"... In the past 20 years there has been treftlendous progress in developing and analyzing parallel algorithftls. Researchers have developed efficient parallel algorithms to solve most problems for which efficient sequential solutions are known. Although some ofthese algorithms are efficient only in a th ..."
Abstract - Cited by 237 (11 self) - Add to MetaCart
In the past 20 years there has been treftlendous progress in developing and analyzing parallel algorithftls. Researchers have developed efficient parallel algorithms to solve most problems for which efficient sequential solutions are known. Although some ofthese algorithms are efficient only in a theoretical framework, many are quite efficient in practice or have key ideas that have been used in efficient implementations. This research on parallel algorithms has not only improved our general understanding ofparallelism but in several cases has led to improvements in sequential algorithms. Unf:ortunately there has been less success in developing good languages f:or prograftlftling parallel algorithftls, particularly languages that are well suited for teaching and prototyping algorithms. There has been a large gap between languages
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...re at each vertex is the same, for example if each vertex has the same number of neighbors. As it turns out, the impasse can be resolved by using randomness to break the symmetry between the vertices =-=[58]-=-. Load balancing: A third use of randomness is load balancing. One way to quickly partition a large number of data items into a collection of approximately evenly sized subsets is to randomly assign e...

The Bit Extraction Problem or t-Resilient Functions

by Benny Chor, Oded Goldreich, Johan Haståd, Joel Freidmann, Steven Rudich, Roman Smolensky , 1985
"... \Gamma We consider the following adversarial situation. Let n, m and t be arbitrary integers, and let f : f0; 1g n 7! f0; 1g m be a function. An adversary, knowing the function f , sets t of the n input bits, while the rest (n \Gamma t input bits) are chosen at random (independently and with un ..."
Abstract - Cited by 171 (11 self) - Add to MetaCart
\Gamma We consider the following adversarial situation. Let n, m and t be arbitrary integers, and let f : f0; 1g n 7! f0; 1g m be a function. An adversary, knowing the function f , sets t of the n input bits, while the rest (n \Gamma t input bits) are chosen at random (independently and with uniform probability distribution). The adversary tries to prevent the outcome of f from being uniformly distributed in f0; 1g m . The question addressed is for what values of n, m and t does the adversary necessarily fail in biasing the outcome of f : f0; 1g n 7! f0; 1g m , when being restricted to set t of the input bits of f . We present various lower and upper bounds on m's allowing an affirmative answer. These bounds are relatively close for t n=3 and for t 2n=3. Our results have applications in the fields of fault-tolerance and cryptography. 1. INTRODUCTION The bit extraction problem formulated above The bit extraction problem was suggested by Brassard and Robert [BRref] and by V...

Approximation Algorithms for Disjoint Paths Problems

by Jon Michael Kleinberg , 1996
"... The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NP-complete problems for w ..."
Abstract - Cited by 166 (0 self) - Add to MetaCart
The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NP-complete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in high-speed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.

What Cannot Be Computed Locally!

by Fabian Kuhn, Thomas Moscibroda, Roger Wattenhofer - In Proceedings of the 23 rd ACM Symposium on the Principles of Distributed Computing (PODC , 2004
"... We give time lower bounds for the distributed approximation of minimum vertex cover (MVC) and related problems such as minimum dominating set (MDS). In k communication rounds, MVC and MDS can only be approximated by factors# /k) and # /k) for some constant c, where n and # denote the number ..."
Abstract - Cited by 137 (27 self) - Add to MetaCart
We give time lower bounds for the distributed approximation of minimum vertex cover (MVC) and related problems such as minimum dominating set (MDS). In k communication rounds, MVC and MDS can only be approximated by factors# /k) and # /k) for some constant c, where n and # denote the number of nodes and the largest degree in the graph. The number of rounds required in order to achieve a constant or even only a polylogarithmic approximation ratio is at log n/ log log n) and#1 #/ log log #). By a simple reduction, the latter lower bounds also hold for the construction of maximal matchings and maximal independent sets.

BoomerAMG: a Parallel Algebraic Multigrid Solver and Preconditioner

by Van Emden Henson, Ulrike Meier Yang - Applied Numerical Mathematics , 2000
"... Driven by the need to solve linear sytems arising from problems posed on extremely large, unstructured grids, there has been a recent resurgence of interest in algebraic multigrid (AMG). AMG is attractive in that it holds out the possibility of multigridlike performance on unstructured grids. The sh ..."
Abstract - Cited by 129 (10 self) - Add to MetaCart
Driven by the need to solve linear sytems arising from problems posed on extremely large, unstructured grids, there has been a recent resurgence of interest in algebraic multigrid (AMG). AMG is attractive in that it holds out the possibility of multigridlike performance on unstructured grids. The sheer size of many modern physics and simulation problems has led to the development of massively parallel computers, and has sparked much research into developing algorithms for them. Parallelizing AMG is a difficult task, however. While much of the AMG method parallelizes readily, the process of coarse-grid selection, in particular, is fundamentally sequential in nature. We have previously introduced a parallel algorithm [7] for the selection of coarsegrid points, based on modifications of certain parallel independent set algorithms and the application of heuristics designed to insure the quality of the coarse grids, and shown results from a prototype serial version of the algorithm. In this pa...
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...describesrst the Cleary-Luby-Jones-Plassman (CLJP) parallel coarsening algorithm. This coarsening was proposed by Cleary [7], and is based on parallel graph partitioning algorithms introduced by Luby =-=[13-=-] and developed by Jones and Plassman [11]. We begin by dening S, the auxiliary in uence matrix: S ij = 8 > > : 1 if j 2 S i ; 0 otherwise: (3) 7 That is, S ij = 1 only if i depends on j. The ith row ...

A Parallel Algorithm for Multilevel Graph Partitioning and Sparse Matrix Ordering

by George Karypis, Vipin Kumar , 1996
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Abstract - Cited by 111 (7 self) - Add to MetaCart
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...llelize algorithms based on depth-first traversal of the graph. Another possibility is to adapt some of the algorithms that have been developed for the PRAM model. In particular the algorithm of Luby =-=[21]-=- for computing the maximal independent set can be used to find a matching. However, parallel formulations of this type of algorithms also have high communication overhead because each processor pi nee...

The algorithmic aspects of the Regularity Lemma

by N. Alon, R. A. Duke, H. Lefmann, V. Rödl, R. Yuster - J. Algorithms , 1994
"... The Regularity Lemma of Szemerédi is a result that asserts that every graph can be par-titioned in a certain regular way. This result has numerous applications, but its known proof is not algorithmic. Here we first demonstrate the computational difficulty of finding a regular partition; we show that ..."
Abstract - Cited by 108 (30 self) - Add to MetaCart
The Regularity Lemma of Szemerédi is a result that asserts that every graph can be par-titioned in a certain regular way. This result has numerous applications, but its known proof is not algorithmic. Here we first demonstrate the computational difficulty of finding a regular partition; we show that deciding if a given partition of an input graph satisfies the properties guaranteed by the lemma is co-NP-complete. However, we also prove that despite this difficulty the lemma can be made constructive; we show how to obtain, for any input graph, a partition with the properties guaranteed by the lemma, efficiently. The desired partition, for an n-vertex graph, can be found in time O(M(n)), where M(n) = O(n 2.376) is the time needed to multiply two n by n matrices with 0, 1-entries over the integers. The algorithm can be parallelized and implemented in NC 1. Besides the curious phenomenon of exhibiting a natural problem in which the search for a solution is easy whereas the decision if a given instance is a solution is difficult (if P and NP differ), our constructive version of the Regularity Lemma supplies efficient sequential and parallel algorithms for many problems, some of which are naturally motivated by the study of various graph embedding and graph coloring problems.
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...′ is of size at least 6m. Such a set can be easily found in NC by applying one of the many known NC algorithms for finding a maximal independent set in an appropriately defined graph (see, e.g., [20],=-=[21]-=-, [4]). (The graph considered here is of course the one whose set of vertices are all possible paths of length 4 with the required endpoints where two are adjacent if they share an internal vertex). O...

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