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Perfectly balanced allocation
 in Proceedings of the 7th International Workshop on Randomization and Approximation Techniques in Computer Science, Princeton, NJ, 2003, Lecture Notes in Comput. Sci. 2764
, 2003
"... Abstract. We investigate randomized processes underlying load balancing based on the multiplechoice paradigm: m balls have to be placed in n bins, and each ball can be placed into one out of 2 randomly selected bins. The aim is to distribute the balls as evenly as possible among the bins. Previousl ..."
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Cited by 23 (1 self)
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Abstract. We investigate randomized processes underlying load balancing based on the multiplechoice paradigm: m balls have to be placed in n bins, and each ball can be placed into one out of 2 randomly selected bins. The aim is to distribute the balls as evenly as possible among the bins. Previously, it was known that a simple process that places the balls one by one in the least loaded bin can achieve a maximum load of m/n + Θ(log log n) with high probability. Furthermore, it was known that it is possible to achieve (with high probability) a maximum load of at most ⌈m/n ⌉ +1using maximum flow computations. In this paper, we extend these results in several aspects. First of all, we show that if m ≥ cn log n for some sufficiently large c, thenaperfect distribution of balls among the bins can be achieved (i.e., the maximum load is ⌈m/n⌉) with high probability. The bound for m is essentially optimal, because it is known that if m ≤ c ′ n log n for some sufficiently small constant c ′ , the best possible maximum load that can be achieved is ⌈m/n ⌉ +1with high probability. Next, we analyze a simple, randomized load balancing process based on a local search paradigm. Our first result here is that this process always converges to a best possible load distribution. Then, we study the convergence speed of the process. We show that if m is sufficiently large compared to n,thenno matter with which ball distribution the system starts, if the imbalance is ∆, then the process needs only ∆·n O(1) steps to reach a perfect distribution, with high probability. We also prove a similar result for m ≈ n, and show that if m = O(n log n / log log n), then an optimal load distribution (which has the maximum load of ⌈m/n ⌉ +1) is reached by the random process after a polynomial number of steps, with high probability.
On the maximum queue length in the supermarket model
, 2004
"... There are n queues, each with a single server. Customers arrive in a Poisson process at rate λn, where0<λ<1. Upon arrival each customer selects d ≥ 2 servers uniformly at random, and joins the queue at a leastloaded server among those chosen. Service times are independent exponentially distrib ..."
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Cited by 16 (2 self)
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There are n queues, each with a single server. Customers arrive in a Poisson process at rate λn, where0<λ<1. Upon arrival each customer selects d ≥ 2 servers uniformly at random, and joins the queue at a leastloaded server among those chosen. Service times are independent exponentially distributed random variables with mean 1. We show that the system is rapidly mixing, and then investigate the maximum length of a queue in the equilibrium distribution. We prove that with probability tending to 1 as n →∞the maximum queue length takes at most two values, which are ln ln n / ln d + O(1). 1. Introduction. We
Randomized Protocols for LowCongestion Circuit Routing in Multistage Interconnection Networks
"... In this paper we study randomized algorithms for circuit switching on multistage networks related to the butterfly. We devise algorithms that route messages by constructing circuits (or paths) for the messages with small congestion, dilation, and setup time. Our algorithms are based on the idea of h ..."
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Cited by 16 (5 self)
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In this paper we study randomized algorithms for circuit switching on multistage networks related to the butterfly. We devise algorithms that route messages by constructing circuits (or paths) for the messages with small congestion, dilation, and setup time. Our algorithms are based on the idea of having each message choose a route from two possibilities, a technique that has previously proven successful in simpler load balancing settings. As an application of our techniques, we propose a novel design for a data server.
Balanced Allocations: The Weighted Case
, 2008
"... We investigate ballsandbins processes where m weighted balls are placed into n bins using the “power of two choices ” paradigm, whereby a ball is inserted into the less loaded of two randomly chosen bins. The case where each of the m balls has unit weight had been studied extensively. In a seminal ..."
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Cited by 15 (3 self)
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We investigate ballsandbins processes where m weighted balls are placed into n bins using the “power of two choices ” paradigm, whereby a ball is inserted into the less loaded of two randomly chosen bins. The case where each of the m balls has unit weight had been studied extensively. In a seminal paper Azar et al. [2] showed that when m = n the most loaded bin has Θ(log log n) balls with high probability. Surprisingly, the gap in load between the heaviest bin and the average bin does not increase with m and was shown by Berenbrink et al. [4] to be Θ(log log n) with high probability for arbitrarily large m. We generalize this result to the weighted case where balls have weights drawn from an arbitrary weight distribution. We show that as long as the weight distribution has finite second moment and satisfies a mild technical condition, the gap between the weight of the heaviest bin and the weight of the average bin is independent of the number balls thrown. This is especially striking when considering heavy tailed distributions such as PowerLaw and LogNormal distributions. In these cases, as more balls are thrown, heavier and heavier weights are encountered. Nevertheless with high probability, the imbalance in the load distribution does not increase. Furthermore, if the fourth moment of the weight distribution is finite, the expected value of the gap is shown to be independent of the number of balls. 1 1
Reducing network congestion and blocking probability through balanced allocation
 in: Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, FOCS
, 1999
"... We compare the performance of a variant of the standard Dynamic Alternative Routing (DAR) technique commonly used in telephone and ATM networks to a path selection algorithm that is based on the balanced allocations principle [4, 18] the Balanced Dynamic Alternative Routing (BDAR) algorithm. While ..."
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Cited by 14 (4 self)
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We compare the performance of a variant of the standard Dynamic Alternative Routing (DAR) technique commonly used in telephone and ATM networks to a path selection algorithm that is based on the balanced allocations principle [4, 18] the Balanced Dynamic Alternative Routing (BDAR) algorithm. While the standard technique checks alternative routes sequentially until available bandwidth is found, the BDAR algorithm compares and chooses the best among a small number of alternatives. We show that, at the expense of a minor increase in routing overhead, the BDAR gives a substantial improvement in network performance in terms of both network congestion and blocking probabilities. 1
Load Balancing in Arbitrary Network Topologies with Stochastic Adversarial Input
 SIAM Journal on Computing
, 2005
"... We study the longterm (steady state) performance of a simple, randomized, local load balancing technique under a broad range of input conditions. We assume a system of n processors connected by an arbitrary network topology. Jobs are placed in the processors by a deterministic or randomized adversa ..."
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Cited by 11 (2 self)
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We study the longterm (steady state) performance of a simple, randomized, local load balancing technique under a broad range of input conditions. We assume a system of n processors connected by an arbitrary network topology. Jobs are placed in the processors by a deterministic or randomized adversary. The adversary knows the current and past load distribution in the network and can use this information to place the new tasks in the processors. A node can execute one job per step, and can also participate in one load balancing operation in which it can move tasks to a direct neighbor in the network. In the protocol we analyze here, a node equalizes its load with a random neighbor in the graph.
Averagecase analyses of first fit and random fit bin packing
 In Proc. Ninth Annual ACMSIAM Symposium on Discrete Algorithms
, 1998
"... ABSTRACT: We prove that the First Fit bin packing algorithm is stable under the input distribution U�k − 2�k � for all k ≥ 3, settling an open question from the recent survey by Coffman, Garey, and Johnson [“Approximation algorithms for bin backing: A survey, ” Approximation algorithms for NPhard p ..."
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Cited by 11 (1 self)
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ABSTRACT: We prove that the First Fit bin packing algorithm is stable under the input distribution U�k − 2�k � for all k ≥ 3, settling an open question from the recent survey by Coffman, Garey, and Johnson [“Approximation algorithms for bin backing: A survey, ” Approximation algorithms for NPhard problems, D. Hochbaum (Editor), PWS, Boston, 1996]. Our proof generalizes the multidimensional Markov chain analysis used by Kenyon, Sinclair, and Rabani to prove that Best Fit is also stable under these distributions [Proc Seventh Annual ACMSIAM Symposium on Discrete Algorithms, 1995, pp. 351–358]. Our proof is motivated by an analysis of Random Fit, a new simple packing algorithm related to First Fit, that is interesting in its own right. We show that Random Fit is stable under the input distributions U�k − 2�k�, as well as present worst case bounds and some results on distributions
Recovery time of dynamic allocation processes
 IN PROCEEDINGS OF THE 10TH ANNUAL ACM SYMPOSIUM ON PARALLEL ALGORITHMS AND ARCHITECTURES, PUERTO VALLARTA, MEXICO, 28 JUNE–2
, 1998
"... Many distributed protocols arising in applications in online load balancing and dynamic resource allocation can be modeled by dynamic allocation processes related to the “balls into bin” problems. Traditionally the main focus of the research on dynamic allocation processes is on verifying whether a ..."
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Cited by 11 (3 self)
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Many distributed protocols arising in applications in online load balancing and dynamic resource allocation can be modeled by dynamic allocation processes related to the “balls into bin” problems. Traditionally the main focus of the research on dynamic allocation processes is on verifying whether a given process is stable, and if so, on analyzing its behavior in the limit (i.e., after sufficiently many steps). Once we know that the process is stable and we know its behavior in the limit, it is natural to analyze its recovery time, which is the time needed by the process to recover from any arbitrarily bad situation and to arrive very closely to a stable (i.e., a typical) state. This investigation is important to provide assurance that even if at some stage the process has reached a highly undesirable state, we can predict with high confidence its behavior after the estimated recovery time. In this paper we present a genera / framework to study the recovery time of discretetime dynamic allocation processes. We model allocation processes by suitably chosen ergodic Markov chains. For a given Markov chain we apply path coupling arguments to bound its convergence rates to the stationary distribution, which directly yields the estimation of the recovery time of the corresponding allocation process. Our coupling approach provides in a relatively simple way an accurate prediction of the recovery time. In particular, we show that our method can be applied to significantly improve estimations of the recovery time for various allocation processes related to allocations of balls into bins, and for the edge orientation problem studied before by Ajtai et al.
Load Profiling for Efficient Route Selection in MultiClass Networks
 IN PROC. IEEE ICNP
, 1997
"... Highspeed networks, such as ATM networks, are expected to support diverse Quality of Service (QoS) constraints, including realtime QoS guarantees. Realtime QoS is required by many applications such as those that involvevoice and video communication. To support such services, routing algorithms ..."
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Cited by 10 (2 self)
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Highspeed networks, such as ATM networks, are expected to support diverse Quality of Service (QoS) constraints, including realtime QoS guarantees. Realtime QoS is required by many applications such as those that involvevoice and video communication. To support such services, routing algorithms that allow applications to reserve the needed bandwidth over a Virtual Circuit (VC) have been proposed. Commonly, these bandwidthreservation algorithms assign VCs to routes using the leastloaded concept, and thus result in balancing the load over the set of all candidate routes.
Asymptotic distributions and chaos for the supermarket model
, 2006
"... In the supermarket model there are n queues, each with a unit rate server. Customers arrive in a Poisson process at rate λn, where 0 < λ < 1. Each customer chooses d ≥ 2 queues uniformly at random, and joins a shortest one. It is known that the equilibrium distribution of a typical queue lengt ..."
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Cited by 9 (1 self)
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In the supermarket model there are n queues, each with a unit rate server. Customers arrive in a Poisson process at rate λn, where 0 < λ < 1. Each customer chooses d ≥ 2 queues uniformly at random, and joins a shortest one. It is known that the equilibrium distribution of a typical queue length converges to a certain explicit limiting distribution as n → ∞. We quantify the rate of convergence by showing that the total variation distance between the equilibrium distribution and the limiting distribution is essentially of order n −1; and we give a corresponding result for systems starting from quite general initial conditions (not in equilibrium). Further, we quantify the result that the systems exhibit chaotic behaviour: we show that the total variation distance between the joint law of a fixed set of queue lengths and the corresponding product law is essentially of order at most n −1.