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148
The Power of Two Choices in Randomized Load Balancing
 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS
, 1996
"... Suppose that n balls are placed into n bins, each ball being placed into a bin chosen independently and uniformly at random. Then, with high probability, the maximum load in any bin is approximately log n log log n . Suppose instead that each ball is placed sequentially into the least full of d ..."
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Cited by 201 (23 self)
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Suppose that n balls are placed into n bins, each ball being placed into a bin chosen independently and uniformly at random. Then, with high probability, the maximum load in any bin is approximately log n log log n . Suppose instead that each ball is placed sequentially into the least full of d bins chosen independently and uniformly at random. It has recently been shown that the maximum load is then only log log n log d +O(1) with high probability. Thus giving each ball two choices instead of just one leads to an exponential improvement in the maximum load. This result demonstrates the power of two choices, and it has several applications to load balancing in distributed systems. In this thesis, we expand upon this result by examining related models and by developing techniques for stu...
Brownian Excursions, Critical Random Graphs and the Multiplicative Coalescent
, 1996
"... Let (B t (s); 0 s ! 1) be reflecting inhomogeneous Brownian motion with drift t \Gamma s at time s, started with B t (0) = 0. Consider the random graph G(n; n \Gamma1 +tn \Gamma4=3 ), whose largest components have size of order n 2=3 . Normalizing by n \Gamma2=3 , the asymptotic joint d ..."
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Cited by 86 (10 self)
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Let (B t (s); 0 s ! 1) be reflecting inhomogeneous Brownian motion with drift t \Gamma s at time s, started with B t (0) = 0. Consider the random graph G(n; n \Gamma1 +tn \Gamma4=3 ), whose largest components have size of order n 2=3 . Normalizing by n \Gamma2=3 , the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of B t (Corollary 2). The dynamics of merging of components as t increases are abstracted to define the multiplicative coalescent process. The states of this process are vectors x of nonnegative real cluster sizes (x i ), and clusters with sizes x i and x j merge at rate x i x j . The multiplicative coalescent is shown to be a Feller process on l 2 . The random graph limit specifies the standard multiplicative coalescent, which starts from infinitesimally small clusters at time \Gamma1: the existence of such a process is not obvious. AMS 1991 subject classifications. 60C05, 60J50, Key words and phras...
Probabilistic and Statistical Properties of Words: An Overview
 Journal of Computational Biology
, 2000
"... In the following, an overview is given on statistical and probabilistic properties of words, as occurring in the analysis of biological sequences. Counts of occurrence, counts of clumps, and renewal counts are distinguished, and exact distributions as well as normal approximations, Poisson process a ..."
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Cited by 84 (1 self)
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In the following, an overview is given on statistical and probabilistic properties of words, as occurring in the analysis of biological sequences. Counts of occurrence, counts of clumps, and renewal counts are distinguished, and exact distributions as well as normal approximations, Poisson process approximations, and compound Poisson approximations are derived. Here, a sequence is modelled as a stationary ergodic Markov chain; a test for determining the appropriate order of the Markov chain is described. The convergence results take the error made by estimating the Markovian transition probabilities into account. The main tools involved are moment generating functions, martingales, Stein’s method, and the ChenStein method. Similar results are given for occurrences of multiple patterns, and, as an example, the problem of unique recoverability of a sequence from SBH chip data is discussed. Special emphasis lies on disentangling the complicated dependence structure between word occurrences, due to selfoverlap as well as due to overlap between words. The results can be used to derive approximate, and conservative, con � dence intervals for tests. Key words: word counts, renewal counts, Markov model, exact distribution, normal approximation, Poisson process approximation, compound Poisson approximation, occurrences of multiple words, sequencing by hybridization, martingales, moment generating functions, Stein’s method, ChenStein method. 1.
Convergence rates of Markov chains
, 1995
"... this paper, we attempt to describe various mathematical techniques which have been used to bound such rates of convergence. In particular, we describe eigenvalue analysis, random walks on groups, coupling, and minorization conditions. Connections are made to modern areas of research wherever possibl ..."
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Cited by 62 (4 self)
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this paper, we attempt to describe various mathematical techniques which have been used to bound such rates of convergence. In particular, we describe eigenvalue analysis, random walks on groups, coupling, and minorization conditions. Connections are made to modern areas of research wherever possible. Elements of linear algebra, probability theory, group theory, and measure theory are used, but efforts are made to keep the presentation elementary and accessible. Acknowledgements. I thank Eric Belsley for comments and corrections, and thank Persi Diaconis for introducing me to this subject and teaching me so much. 1. Introduction and motivation.
Parallel Randomized Load Balancing
 In Symposium on Theory of Computing. ACM
, 1995
"... It is well known that after placing n balls independently and uniformly at random into n bins, the fullest bin holds \Theta(log n= log log n) balls with high probability. Recently, Azar et al. analyzed the following: randomly choose d bins for each ball, and then sequentially place each ball in the ..."
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Cited by 56 (8 self)
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It is well known that after placing n balls independently and uniformly at random into n bins, the fullest bin holds \Theta(log n= log log n) balls with high probability. Recently, Azar et al. analyzed the following: randomly choose d bins for each ball, and then sequentially place each ball in the least full of its chosen bins [2]. They show that the fullest bin contains only log log n= log d + \Theta(1) balls with high probability. We explore extensions of this result to parallel and distributed settings. Our results focus on the tradeoff between the amount of communication and the final load. Given r rounds of communication, we provide lower bounds on the maximum load of \Omega\Gamma r p log n= log log n) for a wide class of strategies. Our results extend to the case where the number of rounds is allowed to grow with n. We then demonstrate parallelizations of the sequential strategy presented in Azar et al. that achieve loads within a constant factor of the lower bound for two ...
Processor Allocation and Checkpoint Interval Selection in Cluster Computing Systems
 Journal of Parallel and Distributed Computing
, 2001
"... Performance prediction of checkpointing systems in the presence of failures is a wellstudied research area. While the literature abounds with performance models of checkpointing systems, none address the issue of selecting runtime parameters other than the optimal checkpointing interval. In parti ..."
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Cited by 33 (0 self)
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Performance prediction of checkpointing systems in the presence of failures is a wellstudied research area. While the literature abounds with performance models of checkpointing systems, none address the issue of selecting runtime parameters other than the optimal checkpointing interval. In particular, the issue of processor allocation is typically ignored. In this paper, we present a performance model for longrunning parallel computations that execute with checkpointing enabled. We then discuss how it is relevant to today's parallel computing environments and software, and present case studies of using the model to select runtime parameters. Keywords: Checkpointing, performance prediction, parameter selection, parallel computation, Markov chain, exponential failure and repair distributions. 1
Entropy and the law of small numbers
 IEEE Trans. Inform. Theory
, 2005
"... Two new informationtheoretic methods are introduced for establishing Poisson approximation inequalities. First, using only elementary informationtheoretic techniques it is shown that, when Sn = �n i=1 Xi is the sum of the (possibly dependent) binary random variables X1, X2,..., Xn, with E(Xi) = p ..."
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Cited by 29 (11 self)
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Two new informationtheoretic methods are introduced for establishing Poisson approximation inequalities. First, using only elementary informationtheoretic techniques it is shown that, when Sn = �n i=1 Xi is the sum of the (possibly dependent) binary random variables X1, X2,..., Xn, with E(Xi) = pi and E(Sn) = λ, then D(PSn�Po(λ)) ≤ n� i=1 p 2 i + � n � i=1 H(Xi) − H(X1, X2,..., Xn), where D(PSn�Po(λ)) is the relative entropy between the distribution of Sn and the Poisson(λ) distribution. The first term in this bound measures the individual smallness of the Xi and the second term measures their dependence. A general method is outlined for obtaining corresponding bounds when approximating the distribution of a sum of general discrete random variables by an infinitely divisible distribution. Second, in the particular case when the Xi are independent, the following sharper bound is established,
Limit Distributions and Random Trees Derived From the Birthday Problem With Unequal Probabilities
, 1998
"... Given an arbitrary distribution on a countable set S consider the number of independent samples required until the first repeated value is seen. Exact and asymptotic formulae are derived for the distribution of this time and of the times until subsequent repeats. Asymptotic properties of the repeat ..."
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Cited by 26 (14 self)
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Given an arbitrary distribution on a countable set S consider the number of independent samples required until the first repeated value is seen. Exact and asymptotic formulae are derived for the distribution of this time and of the times until subsequent repeats. Asymptotic properties of the repeat times are derived by embedding in a Poisson process. In particular, necessary and sufficient conditions for convergence are given and the possible limits explicitly described. Under the same conditions the finite dimensional distributions of the repeat times converge to the arrival times of suitably modified Poisson processes, and random trees derived from the sequence of independent Research supported in part by N.S.F. Grants DMS 9224857, 9404345, 9224868 and 9703691 trials converge in distribution to an inhomogeneous continuum random tree. 1 Introduction Recall the classical birthday problem: given that each day of the year is equally likely as a possible birthday, and that birth...
Computing equilibria in anonymous games
 in 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS
, 2007
"... We present efficient approximation algorithms for finding Nash equilibria in anonymous games, that is, games in which the players utilities, though different, do not differentiate between other players. Our results pertain to such games with many players but few strategies. We show that any such gam ..."
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Cited by 23 (4 self)
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We present efficient approximation algorithms for finding Nash equilibria in anonymous games, that is, games in which the players utilities, though different, do not differentiate between other players. Our results pertain to such games with many players but few strategies. We show that any such game has an approximate pure Nash equilibrium, computable in polynomial time, with approximation O(s 2 λ), where s is the number of strategies and λ is the Lipschitz constant of the utilities. Finally, we show that there is a PTAS for finding an ɛapproximate Nash equilibrium when the number of strategies is two. 1
Probabilistic bounds on the coefficients of polynomials with only real zeros
 J. Combin. Theory Ser. A
, 1997
"... The work of Harper and subsequent authors has shown that nite sequences (a 0;;an) arising from combinatorial problems are often such that the polynomial A(z): = P n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coe cients, Stirling numbers of the rst and sec ..."
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Cited by 20 (0 self)
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The work of Harper and subsequent authors has shown that nite sequences (a 0;;an) arising from combinatorial problems are often such that the polynomial A(z): = P n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coe cients, Stirling numbers of the rst and second kinds, and Eulerian numbers. Assuming the ak are nonnegative, A(1)> 0 and that A(z) is not constant, it is known that A(z) has only real zeros i the normalized sequence (a 0=A(1);;an=A(1)) is the probability distribution of the Research supported in part by N.S.F. Grant MCS9404345 1 number of successes in n independent trials for some sequence of success probabilities. Such sequences (a 0;;an) are also known to be characterized by total positivity of the in nite matrix (ai,j) indexed by nonnegative integers i and j. This papers reviews inequalities and approximations for such sequences, called Polya frequency sequences which follow from their probabilistic representation. In combinatorial examples these inequalities yield a number of improvements of known estimates.