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34
Balanced Allocations
 SIAM Journal on Computing
, 1994
"... Suppose that we sequentially place n balls into n boxes by putting each ball into a randomly chosen box. It is well known that when we are done, the fullest box has with high probability (1 + o(1)) ln n/ ln ln n balls in it. Suppose instead that for each ball we choose two boxes at random and place ..."
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Cited by 326 (8 self)
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Suppose that we sequentially place n balls into n boxes by putting each ball into a randomly chosen box. It is well known that when we are done, the fullest box has with high probability (1 + o(1)) ln n/ ln ln n balls in it. Suppose instead that for each ball we choose two boxes at random and place the ball into the one which is less full at the time of placement. We show that with high probability, the fullest box contains only ln ln n/ ln 2 +O(1) balls  exponentially less than before. Furthermore, we show that a similar gap exists in the infinite process, where at each step one ball, chosen uniformly at random, is deleted, and one ball is added in the manner above. We discuss consequences of this and related theorems for dynamic resource allocation, hashing, and online load balancing.
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 291 (24 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...
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 ..."
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Cited by 137 (6 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.,...
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 63 (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 ...
Adversarial contention resolution for simple channels
 In: 17th Annual Symposium on Parallelism in Algorithms and Architectures
, 2005
"... This paper analyzes the worstcase performance of randomized backoff on simple multipleaccess channels. Most previous analysis of backoff has assumed a statistical arrival model. For batched arrivals, in which all n packets arrive at time 0, we show the following tight highprobability bounds. Rand ..."
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Cited by 49 (1 self)
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This paper analyzes the worstcase performance of randomized backoff on simple multipleaccess channels. Most previous analysis of backoff has assumed a statistical arrival model. For batched arrivals, in which all n packets arrive at time 0, we show the following tight highprobability bounds. Randomized binary exponential backoff has makespan Θ(nlgn), and more generally, for any constant r, rexponential backoff has makespan Θ(nlog lgr n). Quadratic backoff has makespan Θ((n/lg n) 3/2), and more generally, for r> 1, rpolynomial backoff has makespan Θ((n/lg n) 1+1/r). Thus, for batched inputs, both exponential and polynomial backoff are highly sensitive to backoff constants. We exhibit a monotone superpolynomial subexponential backoff algorithm, called loglogiterated backoff, that achieves makespan Θ(nlg lgn/lg lglgn). We provide a matching lower bound showing that this strategy is optimal among all monotone backoff algorithms. Of independent interest is that this lower bound was proved with a delay sequence argument. In the adversarialqueuing model, we present the following stability and instability results for exponential backoff and loglogiterated backoff. Given a (λ,T)stream, in which at most n = λT packets arrive in any interval of size T, exponential backoff is stable for arrival rates of λ = O(1/lgn) and unstable for arrival rates of λ = Ω(lglgn/lg n); loglogiterated backoff is stable for arrival rates of λ = O(1/(lg lgnlgn)) and unstable for arrival rates of λ = Ω(1/lg n). Our instability results show that bursty input is close to being worstcase for exponential backoff and variants and that even small bursts can create instabilities in the channel.
Load Balancing and Density Dependent Jump Markov Processes (Extended Abstract)
 IN PROCEEDINGS OF THE 37TH IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 1996
"... We provide a new approach for analyzing both static and dynamic randomized load balancing strategies. We demonstrate the approach by providing the first analysis of the following model: customers arrive as a Poisson stream of rate n, ! 1, at a collection of n servers. Each customer chooses some co ..."
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Cited by 48 (12 self)
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We provide a new approach for analyzing both static and dynamic randomized load balancing strategies. We demonstrate the approach by providing the first analysis of the following model: customers arrive as a Poisson stream of rate n, ! 1, at a collection of n servers. Each customer chooses some constant d servers independently and uniformly at random from the n servers, and waits for service at the one with the fewest customers. Customers are served according to the firstin firstout (FIFO) protocol, and the service time for a customer is exponentially distributed with mean 1. We call this problem the supermarket model. We wish to know how the system behaves, and in particular we are interested the expected time a customer spends in the system in equilibrium. The model provides a good abstraction of a simple, efficient load balancing scheme in the setting where jobs...
Stochastic Contention Resolution With Short Delays
 In Proc. 27th ACM Symp. on Theory of Computing
"... We study contention resolution protocols under a stochastic model of continuous request generation from a set of contenders. The performance of such a protocol is characterized by two parameters: the maximum arrival rate for which the protocol is stable and the expected delay of a request from ar ..."
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Cited by 42 (1 self)
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We study contention resolution protocols under a stochastic model of continuous request generation from a set of contenders. The performance of such a protocol is characterized by two parameters: the maximum arrival rate for which the protocol is stable and the expected delay of a request from arrival to service. Known solutions are either unstable for any constant injection rate, or have polynomial (in the number of contenders) expected delay. Our main contribution is a protocol that is stable for a constant injection rate, while achieving logarithmic expected delay. We extend our results to the case of multiple servers, with each request being targeted for a specific server. This is related to the optically connected parallel computer (or OCPC) model. Finally, we prove a lower bound showing that long delays are inevitable in a class of protocols including backoffstyle protocols, if the arrival rate is large enough (but still smaller than 1). 1. Introduction The subject...
Contention Resolution with Constant Expected Delay
"... We study contention resolution problem in a multipleaccess channel such as the Ethernet... ..."
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Cited by 29 (3 self)
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We study contention resolution problem in a multipleaccess channel such as the Ethernet...
Parallel Balanced Allocations
 IN PROCEEDINGS OF THE 8TH ANNUAL ACM SYMPOSIUM ON PARALLEL ALGORITHMS AND ARCHITECTURES
, 1996
"... We study the well known problem of throwing m balls into n bins. If each ball in the sequential game is allowed to select more than one bin, the maximum load of the bins can be exponentially reduced compared to the `classical balls into bins' game. We consider a static and a dynamic variant of ..."
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Cited by 24 (1 self)
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We study the well known problem of throwing m balls into n bins. If each ball in the sequential game is allowed to select more than one bin, the maximum load of the bins can be exponentially reduced compared to the `classical balls into bins' game. We consider a static and a dynamic variant of a randomized parallel allocation where each ball can choose a constant number of bins. All results hold with high probability. In the static case all m balls arrive at the same time. We analyze for m = n a very simple optimal class of protocols achieving maximum load O i r q log n log log n j if r rounds of communication are allowed. This matches the lower bound of [ACMR95]. Furthermore, we generalize the protocols to the case of m ? n balls. An optimal load of O(m=n) can be achieved using log log n log(m=n) rounds of communication. Hence, for m = n log log n log log log n balls this slackness allows to hide the amount of communication. In the `classical balls into bins' game this op...
Approximation Algorithms Via Randomized Rounding: A Survey
 Series in Advanced Topics in Mathematics, Polish Scientific Publishers PWN
, 1999
"... Approximation algorithms provide a natural way to approach computationally hard problems. There are currently many known paradigms in this area, including greedy algorithms, primaldual methods, methods based on mathematical programming (linear and semidefinite programming in particular), local i ..."
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Cited by 20 (2 self)
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Approximation algorithms provide a natural way to approach computationally hard problems. There are currently many known paradigms in this area, including greedy algorithms, primaldual methods, methods based on mathematical programming (linear and semidefinite programming in particular), local improvement, and "low distortion" embeddings of general metric spaces into special families of metric spaces. Randomization is a useful ingredient in many of these approaches, and particularly so in the form of randomized rounding of a suitable relaxation of a given problem. We survey this technique here, with a focus on correlation inequalities and their applications.