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10
Randomness is Linear in Space
 Journal of Computer and System Sciences
, 1993
"... We show that any randomized algorithm that runs in space S and time T and uses poly(S) random bits can be simulated using only O(S) random bits in space S and time T poly(S). A deterministic simulation in space S follows. Of independent interest is our main technical tool: a procedure which extracts ..."
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Cited by 229 (20 self)
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We show that any randomized algorithm that runs in space S and time T and uses poly(S) random bits can be simulated using only O(S) random bits in space S and time T poly(S). A deterministic simulation in space S follows. Of independent interest is our main technical tool: a procedure which extracts randomness from a defective random source using a small additional number of truly random bits. 1
Dispersers, Deterministic Amplification, and Weak Random Sources.
, 1989
"... We use a certain type of expanding bipartite graphs, called disperser graphs, to design procedures for picking highly correlated samples from a finite set, with the property that the probability of hitting any sufficiently large subset is high. These procedures require a relatively small number of r ..."
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Cited by 93 (11 self)
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We use a certain type of expanding bipartite graphs, called disperser graphs, to design procedures for picking highly correlated samples from a finite set, with the property that the probability of hitting any sufficiently large subset is high. These procedures require a relatively small number of random bits and are robust with respect to the quality of the random bits. Using these sampling procedures to sample random inputs of polynomial time probabilistic algorithms, we can simulate the performance of some probabilistic algorithms with less random bits or with low quality random bits. We obtain the following results: 1. The error probability of an RP or BPP algorithm that operates with a constant error bound and requires n random bits, can be made exponentially small (i.e. 2 \Gamman ), with only (3 + ffl)n random bits, as opposed to standard amplification techniques that require \Omega\Gamma n 2 ) random bits for the same task. This result is nearly optimal, since the informati...
A New Rounding Procedure for the Assignment Problem with Applications to Dense Graph Arrangement Problems
, 2001
"... We present a randomized procedure for rounding fractional perfect matchings to (integral) matchings. If the original fractional matching satis es any linear inequality, then with high probability, the new matching satis es that linear inequality in an approximate sense. This extends the wellkn ..."
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Cited by 77 (3 self)
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We present a randomized procedure for rounding fractional perfect matchings to (integral) matchings. If the original fractional matching satis es any linear inequality, then with high probability, the new matching satis es that linear inequality in an approximate sense. This extends the wellknown LP rounding procedure of Raghavan and Thompson, which is usually used to round fractional solutions of linear programs.
A parallel algorithmic version of the local lemma
, 1991
"... The Lovász Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for ..."
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Cited by 60 (10 self)
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The Lovász Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for converting some of these existence proofs into efficient algorithmic procedures, at the cost of loosing a little in the estimates. His method does not seem to be parallelizable. Here we modify his technique and achieve an algorithmic version that can be parallelized, thus obtaining deterministic NC 1 algorithms for several interesting algorithmic problems.
Randomized Distributed Edge Coloring via an Extension of the ChernoffHoeffding Bounds
 SIAM J. Comput
, 1997
"... . Certain types of routing, scheduling, and resourceallocation problems in a distributed setting can be modeled as edgecoloring problems. We present fast and simple randomized algorithms for edge coloring a graph in the synchronous distributed pointtopoint model of computation. Our algorithms co ..."
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Cited by 56 (9 self)
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. Certain types of routing, scheduling, and resourceallocation problems in a distributed setting can be modeled as edgecoloring problems. We present fast and simple randomized algorithms for edge coloring a graph in the synchronous distributed pointtopoint model of computation. Our algorithms compute an edge coloring of a graph G with n nodes and maximum degree # with at most 1.6# +O(log 1+# n) colors with high probability (arbitrarily close to 1) for any fixed #>0; they run in polylogarithmic time. The upper bound on the number of colors improves upon the (2#  1)coloring achievable by a simple reduction to vertex coloring. To analyze the performance of our algorithms, we introduce new techniques for proving upper bounds on the tail probabilities of certain random variables. The Cherno#Hoe#ding bounds are fundamental tools that are used very frequently in estimating tail probabilities. However, they assume stochastic independence among certain random variables, which may n...
An Extension of the Lovász Local Lemma, and its Applications to Integer Programming
 In Proceedings of the 7th Annual ACMSIAM Symposium on Discrete Algorithms
, 1996
"... The Lov'asz Local Lemma (LLL) is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mea ..."
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Cited by 31 (6 self)
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The Lov'asz Local Lemma (LLL) is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mean. We consider three classes of NPhard integer programs: minimax, packing, and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan & Thompson to derive good approximation algorithms for such problems. We use our extended LLL to prove that randomized rounding produces, with nonzero probability, much better feasible solutions than known before, if the constraint matrices of these integer programs are sparse (e.g., VLSI routing using short paths, problems on hypergraphs with small dimension/degree). We also generalize the method of pessimistic estimators due to Raghavan, to constructivize our packing and covering results. 1
Uniform Generation of NPwitnesses using an NPoracle
 Information and Computation
, 1997
"... A Uniform Generation procedure for NP is an algorithm which given any input in a fixed NPlanguage, outputs a uniformly distributed NPwitness for membership of the input in the language. We present a Uniform Generation procedure for NP that runs in probabilistic polynomialtime with an NPoracle. T ..."
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Cited by 24 (1 self)
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A Uniform Generation procedure for NP is an algorithm which given any input in a fixed NPlanguage, outputs a uniformly distributed NPwitness for membership of the input in the language. We present a Uniform Generation procedure for NP that runs in probabilistic polynomialtime with an NPoracle. This improves upon results of Jerrum, Valiant and Vazirani, which either require a \Sigma P 2 oracle or obtain only almost uniform generation. Our procedure utilizes ideas originating in the works of Sipser, Stockmeyer, and Jerrum, Valiant and Vazirani. Dept. of Computer Science & Engineering, University of California at San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA. EMail: mihir@cs.ucsd.edu. URL: http://wwwcse.ucsd.edu/users/mihir. Supported in part by NSF CAREER Award CCR9624439 and a 1996 Packard Foundation Fellowship in Science and Engineering. y Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel. EMail: oded@wis...
Approximating Scheduling Problems in Parallel
 In EuroPAR 97
, 1997
"... . We show how to approximate in NC the problem of Scheduling Unrelated Parallel Machines, for a fixed number of machines. We develop a (2 + ")approximate parallel algorithm for the problem. Our approach shows how to relate the linear program obtained by relaxing the integer programming formulation ..."
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Cited by 1 (0 self)
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. We show how to approximate in NC the problem of Scheduling Unrelated Parallel Machines, for a fixed number of machines. We develop a (2 + ")approximate parallel algorithm for the problem. Our approach shows how to relate the linear program obtained by relaxing the integer programming formulation of the problem with a linear program formulation that is positive and in the packing/covering form. The relationship established enables us to transfer approximate fractional solutions from the later formulation that is known to be approximable in NC. Then, we show how to obtain an integer approximate solution, i.e. a schedule, from the fractional one, using the randomized rounding technique. Finally, we show that the same technique can be applied to the General Assignment Problem of fixed number of machines and a given makespan T , thus yielding a schedule whose cost is at most (2+ ") times the minimum cost and has makespan at most 2T . 1 Introduction Linear Programming plays an important ...
Analysis of Parallel Algorithms for Finding A Maximal Independent Set in A Random Hypergraph
, 1996
"... It is well known [9] that finding a maximal independent set in a graph is in class NC, and [10] that finding a maximal independent set in a hypergraph with fixed dimension is in RNC. It is not known whether this latter problem remains in NC when the dimension is part of the input. We will study the ..."
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Cited by 1 (0 self)
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It is well known [9] that finding a maximal independent set in a graph is in class NC, and [10] that finding a maximal independent set in a hypergraph with fixed dimension is in RNC. It is not known whether this latter problem remains in NC when the dimension is part of the input. We will study the problem when the problem instances are randomly chosen. It was shown in [6] that the expected running time of a simple parallel algorithm for finding the lexicographically first maximal independent set (lfmis) in a random simple graph is logarithmic in the input size. In this paper, we will prove a generalization of this result. We show that if a random kuniform hypergraph has vertex set f1; 2; : : : ; ng and its edges are chosen independently with probability p from the set of \Gamma n k \Delta possible edges, then our algorithm finds the lfmis Department of Mathematics, Carnegie Mellon University.Pittsburgh, PA 15213 y Department of Mathematics, Carnegie Mellon University. Pittsb...
Parallel Algorithms via the Probabilistic Method
"... We give an introduction to the design of parallel algorithms with the probabilistic method. Algorithms of this kind usually possess a randomized sequential counterpart. Parallelization of such algorithms is inherently linked with derandomization, either ..."
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We give an introduction to the design of parallel algorithms with the probabilistic method. Algorithms of this kind usually possess a randomized sequential counterpart. Parallelization of such algorithms is inherently linked with derandomization, either