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The Power of Two Random Choices: A Survey of Techniques and Results
 in Handbook of Randomized Computing
, 2000
"... ITo motivate this survey, we begin with a simple problem that demonstrates a powerful fundamental idea. Suppose that n balls are thrown into n bins, with each ball choosing a bin independently and uniformly at random. Then the maximum load, or the largest number of balls in any bin, is approximately ..."
Abstract

Cited by 98 (2 self)
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ITo motivate this survey, we begin with a simple problem that demonstrates a powerful fundamental idea. Suppose that n balls are thrown into n bins, with each ball choosing a bin independently and uniformly at random. Then the maximum load, or the largest number of balls in any bin, is approximately log n= log log n with high probability. Now suppose instead that the balls are placed sequentially, and each ball is placed in the least loaded of d 2 bins chosen independently and uniformly at random. Azar, Broder, Karlin, and Upfal showed that in this case, the maximum load is log log n= log d + (1) with high probability [ABKU99]. The important implication of this result is that even a small amount of choice can lead to drastically different results in load balancing. Indeed, having just two random choices (i.e.,...
Distributed selfish load balancing
, 2006
"... Suppose that a set of m tasks are to be shared as equally as possible amongst a set of n resources. A gametheoretic mechanism to find a suitable allocation is to associate each task with a “selfish agent”, and require each agent to select a resource, with the cost of a resource being the number of ..."
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Cited by 29 (1 self)
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Suppose that a set of m tasks are to be shared as equally as possible amongst a set of n resources. A gametheoretic mechanism to find a suitable allocation is to associate each task with a “selfish agent”, and require each agent to select a resource, with the cost of a resource being the number of agents to select it. Agents would then be expected to migrate from overloaded to underloaded resources, until the allocation becomes balanced. Recent work has studied the question of how this can take place within a distributed setting in which agents migrate selfishly without any centralized control. In this paper we discuss a natural protocol for the agents which combines the following desirable features: It can be implemented in a strongly distributed setting, uses no central control, and has good convergence properties. For m ≫ n, the system becomes approximately balanced (an ǫNash equilibrium) in expected time O(log log m). We show using a martingale technique that the process converges to a perfectly balanced allocation in expected time O(log log m + n 4). We also give a lower bound of Ω(max{loglog m, n}) for the convergence time.
Allocating Weighted Jobs in Parallel
, 1997
"... It is well known that after placing m n balls independently and uniformly at random (i.u.r.) into n bins, the fullest bin contains \Theta(log n= log log n+ m n ) balls, with high probability. It is also known (see [Ste96]) that a maximum load of O \Gamma m n \Delta can be obtained for all m n ..."
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Cited by 12 (4 self)
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It is well known that after placing m n balls independently and uniformly at random (i.u.r.) into n bins, the fullest bin contains \Theta(log n= log log n+ m n ) balls, with high probability. It is also known (see [Ste96]) that a maximum load of O \Gamma m n \Delta can be obtained for all m n if a ball is allocated in one (suitably chosen) of two (i.u.r.) bins. Stemann ([Ste96]) shows that r communication rounds suffice to guarantee a maximum load of maxf r p log n; O \Gamma m n \Delta g, with high probability. Adler et al. have shown in [ACMR95] that Stemanns protocol is optimal for constant r. In this paper we extend the above results in two directions: We generalize the lower bound to arbitrary r log log n. This implies that the result of Stemanns protocol is optimal for all r. Our main result is a generalization of Stemanns upper bound to weighted jobs: Let W A (W M ) denote the average (maximum) weight of the balls. Further let \Delta = W A =W M . Note that...
On the power of two choices: balls and bins in continuous time
, 2005
"... Suppose that there are n bins, and balls arrive in a Poisson process at rate λn, whereλ>0 is a constant. Upon arrival, each ball chooses a fixed number d of random bins, and is placed into one with least load. Balls have independent exponential lifetimes with unit mean. We show that the system conve ..."
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Cited by 11 (2 self)
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Suppose that there are n bins, and balls arrive in a Poisson process at rate λn, whereλ>0 is a constant. Upon arrival, each ball chooses a fixed number d of random bins, and is placed into one with least load. Balls have independent exponential lifetimes with unit mean. We show that the system converges rapidly to its equilibrium distribution; and when d ≥ 2, there is an integervalued function md(n) = ln ln n/ln d + O(1) such that, in the equilibrium distribution, the maximum load of a bin is concentrated on the two values md(n) and md(n) − 1, with probability tending to 1, as n →∞. We show also that the maximum load usually does not vary by more than a constant amount from ln ln n/ln d, even over quite long periods of time. 1. Introduction. Ballsandbins processes have been useful for modeling and analyzing a wide range of problems, in discrete mathematics, computer science and communication theory, and, in particular, for problems which involve load sharing, see, for example, [4, 5, 12, 15–17, 22]. Here is one central result, from [3]. Let d be a fixed integer at least 2. Suppose that there are n bins, and n balls arrive
Switching Networks for Generating Random Permutations
 Advances in Switching Networks
, 2001
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Simple Competitive Request Scheduling Strategies
 in 11th ACM Symposium on Parallel Architectures and Algorithms
, 1999
"... In this paper we study the problem of scheduling realtime requests in distributed data servers. We assume the time to be divided into time steps of equal length called rounds. During every round a set of requests arrives at the system, and every resource is able to fulfill one request per round. Ev ..."
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Cited by 2 (0 self)
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In this paper we study the problem of scheduling realtime requests in distributed data servers. We assume the time to be divided into time steps of equal length called rounds. During every round a set of requests arrives at the system, and every resource is able to fulfill one request per round. Every request specifies two (distinct) resources and requires to get access to one of them. Furthermore, every request has a deadline of d, i.e. a request that arrives in round t has to be fulfilled during round t +d 1 at the latest. The number of requests which arrive during some round and the two alternative resources of every request are selected by an adversary. The goal is to maximize the number of requests that are fulfilled before their deadlines expire. We examine the scheduling problem in an online setting, i.e. new requests continuously arrive at the system, and we have to determine online an assignment of the requests to the resources in such a way that every resource has to fulfil...
Infinite Parallel Job Allocation (Extended Abstract)
, 2000
"... ) Petra Berenbrink Dept. of Mathematics & Computer Science Paderborn University D33095 Paderborn, Germany pebe@unipaderborn.de Artur Czumaj y Department of Computer and Information Science New Jersey Institute of Technology University Heights, Newark, NJ 071021982, USA czumaj@cis.njit. ..."
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) Petra Berenbrink Dept. of Mathematics & Computer Science Paderborn University D33095 Paderborn, Germany pebe@unipaderborn.de Artur Czumaj y Department of Computer and Information Science New Jersey Institute of Technology University Heights, Newark, NJ 071021982, USA czumaj@cis.njit.edu Tom Friedetzky Institut fur Informatik Technische Universitat Munchen D80290 Munchen, Germany friedetz@informatik.tumuenchen.de Nikita D. Vvedenskaya Institute of Information Transmission Problems Russian Academy of Science Moscow 101447, Russia ndv@iitp.ru Abstract In recent years, the task of allocating jobs to servers has been studied with the \balls and bins" abstraction. Results in this area exploit the large decrease in maximum load that can be achieved by allowing each job (ball) a little freedom in choosing its destination server (bin). In this paper we examine an innite and parallel allocation process (see [ABS98]) which is related to the \balls and bins" abs...
1.1. Problem of Random Permuting
, 1999
"... ABSTRACT: We consider the problem of generating permutations almost uniformly at random in distributed and parallel systems. We propose a simple distributed scheme for permuting at random, which we call distributed mixing, and provide its precise stochastic analysis. Our main result is that distribu ..."
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ABSTRACT: We consider the problem of generating permutations almost uniformly at random in distributed and parallel systems. We propose a simple distributed scheme for permuting at random, which we call distributed mixing, and provide its precise stochastic analysis. Our main result is that distributed mixing needs ��log n � simple pointtopoint communication rounds to generate a permutation almost uniformly at random. We further apply distributed mixing to design very fast parallel algorithms for OCPC and QRQW parallel computers (with runtimes ��log log n � and � � √ log n� � respectively). Our analysis of distributed mixing is based on the analysis of the mixing time of the Markov chain governing the process. The main technical tool developed in the paper is a novel method of analyzing convergence of Markov chains. Our method, called delayed path coupling, is a refinement of the classical coupling technique and the path coupling technique of Bubley and Dyer, and its main, novel feature is the use of possible nonMarkovian coupling. © 2000 John Wiley & Sons, Inc.