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41
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
Network Decomposition and Locality in Distributed Computation (Extended Abstract)
, 1989
"... ) Baruch Awerbuch Department of Mathematics and Laboratory for Computer Science M.I.T. Cambridge, MA 02139 Andrew V. Goldberg y Department of Computer Science Stanford University Stanford, CA 94305 Michael Luby z International Computer Science Institute Berkeley, CA 94704 Serge A. Plo ..."
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Cited by 84 (5 self)
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) Baruch Awerbuch Department of Mathematics and Laboratory for Computer Science M.I.T. Cambridge, MA 02139 Andrew V. Goldberg y Department of Computer Science Stanford University Stanford, CA 94305 Michael Luby z International Computer Science Institute Berkeley, CA 94704 Serge A. Plotkin x Department of Computer Science Stanford University Stanford, CA 94305 May 1989 Abstract We introduce a concept of network decomposition, the essence of which is to partition an arbitrary graph into smalldiameter connected components, such that the graph created by contracting each component into a single node has low chromatic number. We present an efficient distributed algorithm for constructing such a decomposition, and demonstrate its use for design of efficient distributed algorithms. Our method yields new deterministic distributed algorithms for finding a maximal independent set and for (\Delta + 1)coloring of graphs with maximum degree \Delta. These algorithms run...
Derandomization, witnesses for Boolean matrix multiplication and construction of perfect hash functions
 Algorithmica
, 1996
"... Small sample spaces with almost independent random variables are applied to design efficient sequential deterministic algorithms for two problems. The first algorithm, motivated by the attempt to design efficient algorithms for the All Pairs Shortest Path problem using fast matrix multiplication, so ..."
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Cited by 62 (6 self)
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Small sample spaces with almost independent random variables are applied to design efficient sequential deterministic algorithms for two problems. The first algorithm, motivated by the attempt to design efficient algorithms for the All Pairs Shortest Path problem using fast matrix multiplication, solves the problem of computing witnesses for the Boolean product of two matrices. That is, if A and B are two n by n matrices, and C = AB is their Boolean product, the algorithm finds for every entry Cij = 1 a witness: an index k so that Aik = Bkj = 1. Its running time exceeds that of computing the product of two n by n matrices with small integer entries by a polylogarithmic factor. The second algorithm is a nearly linear time deterministic procedure for constructing a perfect hash function for a given nsubset of {1,..., m}.
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...
The probabilistic method yields deterministic parallel algorithms
 Journal of Computer and System Sciences
, 1989
"... We present a technique for converting RNC algorithms into NC algorithms. Our approach is based on a parallel implementation of the method of conditional probabilities. This method was used to convert probabilistic proofs of existence of combinatorial structures into polynomial time deterministic alg ..."
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Cited by 53 (7 self)
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We present a technique for converting RNC algorithms into NC algorithms. Our approach is based on a parallel implementation of the method of conditional probabilities. This method was used to convert probabilistic proofs of existence of combinatorial structures into polynomial time deterministic algorithms. It has the apparent drawback of being extremely sequential in nature. We show certain general conditions under which it is possible to use this technique for devising deterministic parallel algorithms. We use our technique to devise an NC algorithm for the set balancing problem. This problem turns out to be a useful tool for parallel algorithms. Using our derandomization method and the set balancing algorithm, we provide an NC algorithm for the lattice approximation problem. We also use the lattice approximation problem to bootstrap the set balancing algorithm, and the result is a more processor efficient algorithm. The set balancing algorithm also yields an NC algorithm for nearoptimal edge coloring of simple graphs. Our methods also extend to the parallelization of various algorithms in computational geometry that rely upon the random sampling technique of Clarkson. Finally, our methods apply to constructing certain combinatorial structures, e.g. ...
DRAND: Distributed randomized TDMA scheduling for wireless ad hoc networks
 in MobiHoc
, 2006
"... This paper presents a distributed implementation of RAND, a randomized time slot scheduling algorithm, called DRAND. DRAND runs in O(δ) time and message complexity where δ is the maximum size of a twohop neighborhood in a wireless network while message complexity remains O(δ), assuming that message ..."
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Cited by 49 (1 self)
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This paper presents a distributed implementation of RAND, a randomized time slot scheduling algorithm, called DRAND. DRAND runs in O(δ) time and message complexity where δ is the maximum size of a twohop neighborhood in a wireless network while message complexity remains O(δ), assuming that message delays can be bounded by an unknown constant. DRAND is the first fully distributed version of RAND. The algorithm is suitable for a wireless network where most nodes do not move, such as wireless mesh networks and wireless sensor networks. We implement the algorithm in TinyOS and demonstrate its performance in a real testbed of Mica2 nodes. The algorithm does not require any time synchronization and is shown to be effective in adapting to local topology changes without incurring global overhead in the scheduling. Because of these features, it can also be used even for other scheduling problems such as frequency or code scheduling (for FDMA or CDMA) or local identifier assignment for wireless networks where time synchronization is not enforced.
Splitters and NearOptimal Derandomization
, 1995
"... We present a fairly general method for finding deterministic constructions obeying what we call k restrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)universal sets (a collection of binary vectors of leng ..."
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Cited by 39 (2 self)
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We present a fairly general method for finding deterministic constructions obeying what we call k restrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)universal sets (a collection of binary vectors of length n such that for any subset of size k of the indices, all 2 configurations appear) and families of perfect hash functions. The nearoptimal constructions of these objects imply the very efficient derandomization of algorithms in learning, of fixedsubgraph finding algorithms, and of near optimal \Sigma\Pi\Sigma threshold formulae. In addition, they derandomize the reduction showing the hardness of approximation of set cover. They also yield deterministic constructions for a localcoloring protocol, and for exhaustive testing of circuits.
Improved Algorithms via Approximations of Probability Distributions
 Journal of Computer and System Sciences
, 1997
"... We present two techniques for approximating probability distributions. The first is a simple method for constructing the smallbias probability spaces introduced by Naor & Naor. We show how to efficiently combine this construction with the method of conditional probabilities to yield improved NC alg ..."
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Cited by 24 (2 self)
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We present two techniques for approximating probability distributions. The first is a simple method for constructing the smallbias probability spaces introduced by Naor & Naor. We show how to efficiently combine this construction with the method of conditional probabilities to yield improved NC algorithms for many problems such as set discrepancy, finding large cuts in graphs, finding large acyclic subgraphs etc. The second is a construction of small probability spaces approximating general independent distributions, which is of smaller size than the constructions of Even, Goldreich, Luby, Nisan & Velickovi'c. Such approximations are useful, e.g., for the derandomization of certain randomized algorithms. Keywords. Derandomization, parallel algorithms, discrepancy, graph coloring, small sample spaces, explicit constructions. 1 Introduction Derandomization, the development of general tools to derive efficient deterministic algorithms from their randomized counterparts, has blossomed ...
Constructing a Maximal Independent Set in Parallel
 SIAM J. Disc. Math
, 1989
"... f a The problem of constructing in parallel a maximal independent set o given graph is considered. A new deterministic NC algorithm imple  t mented in the EREW PRAM model is presented. On graphs with n ver ices and m edges, it uses O ((n +m )/logn ) processors and runs in O (log n ) 3  c time. T ..."
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Cited by 21 (1 self)
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f a The problem of constructing in parallel a maximal independent set o given graph is considered. A new deterministic NC algorithm imple  t mented in the EREW PRAM model is presented. On graphs with n ver ices and m edges, it uses O ((n +m )/logn ) processors and runs in O (log n ) 3  c time. This reduces by a factor of logn both the running time and the pro essor count of the previously fastest deterministic algorithm which solves the problem using a linear number of processors. Key words: parallel computation, NC, graph, maximal independent set, 1 deterministic. . Introduction The problem of constructing in parallel a maximal independent set of a given graph, t MIS , has been investigated in several recent papers. Karp and Wigderson proved in [KW] hat the problem is in NC . Their algorithm finds a maximal independent set of an n  vertex graph in O (log n ) time and uses O (n /log n ) processors. In successive papers, the 4 3 3  s authors proposed algorithms which either...
On Construction of kwise Independent Random Variables
, 1994
"... A 01 probability space is a probability space(\Omega ; 2\Omega ; P ), where the sample space\Omega ` f0; 1g n for some n. A probability space is kwise independent if, when Y i is defined to be the ith coordinate of the random nvector, then any subset of k of the Y i 's is (mutually) indepen ..."
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Cited by 18 (1 self)
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A 01 probability space is a probability space(\Omega ; 2\Omega ; P ), where the sample space\Omega ` f0; 1g n for some n. A probability space is kwise independent if, when Y i is defined to be the ith coordinate of the random nvector, then any subset of k of the Y i 's is (mutually) independent, and it is said to be a probability space for p1 ; p2 ; :::; pn if P [Y i = 1] = p i . We study constructions of kwise independent 01 probability spaces in which the p i 's are arbitrary. It was known that for any p1 ; p2 ; :::; pn , a kwise independent probability space of size m(n;k) = \Gamma n k \Delta + \Gamma n k\Gamma1 \Delta + \Gamma n k\Gamma2 \Delta + \Delta \Delta \Delta + \Gamma n 0 \Delta always exists. We prove that for some p1 ; p2 ; :::; pn 2 [0; 1], m(n;k) is a lower bound on the size of any kwise independent 01 probability space. For each fixed k, we prove that every kwise independent 01 probability space for all p i = k=n has size\Omega\Gamma n ...