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16
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 122 (4 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.,...
On Balls and Bins with Deletions
 In Proc. of the RANDOM'98
, 1998
"... Microsystems. The views and conclusions contained here are those of the authors and should not be interpreted as necessarily representing the official policies or ..."
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Cited by 20 (1 self)
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Microsystems. The views and conclusions contained here are those of the authors and should not be interpreted as necessarily representing the official policies or
Tight Bounds for Parallel Randomized Load Balancing
 Computing Research Repository
, 1992
"... We explore the fundamental limits of distributed ballsintobins algorithms, i.e., algorithms where balls act in parallel, as separate agents. This problem was introduced by Adler et al., who showed that nonadaptive and symmetric algorithms cannot reliably perform better than a maximum bin load of Θ ..."
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Cited by 18 (7 self)
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We explore the fundamental limits of distributed ballsintobins algorithms, i.e., algorithms where balls act in parallel, as separate agents. This problem was introduced by Adler et al., who showed that nonadaptive and symmetric algorithms cannot reliably perform better than a maximum bin load of Θ(loglogn/logloglogn) within the same number of rounds. We present an adaptive symmetric algorithm that achieves a bin load of two in log ∗ n + O(1) communication rounds using O(n) messages in total. Moreover, larger bin loads can be traded in for smaller time complexities. We prove a matching lower bound of (1−o(1))log ∗ n on the time complexity of symmetric algorithms that guarantee small bin loads at an asymptotically optimal message complexity of O(n). The essential preconditions of the proof are (i) a limit of O(n) on the total number of messages sent by the algorithm and (ii) anonymity of bins, i.e., the port numberings of balls are not globally consistent. In order to show that our technique yields indeed tight bounds, we provide for each assumption an algorithm violating it, in turn achieving a constant maximum bin load in constant time. As an application, we consider the following problem. Given a fully connected graph of n nodes, where each node needs to send and receive up to n messages, and in each round each node may send one message over each link, deliver all messages as quickly as possible to their destinations. We give a simple and robust algorithm of time complexity O(log ∗ n) for this task and provide a generalization to the case where all nodes initially hold arbitrary sets of messages. Completing the picture, we give a less practical, but asymptotically optimal algorithm terminating within O(1) rounds. All these bounds hold with high probability.
Analyzing an Infinite Parallel Job Allocation Process
"... 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 very small amount of choice in choosing its destination serve ..."
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Cited by 14 (8 self)
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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 very small amount of choice in choosing its destination server (bin). The scenarios considered can be divided into two categories: sequential, where each job can be placed at a server before the next job arrives, and parallel, where the jobs arrive in large batches that must be dealt with simultaneously. Another, orthogonal, classification of load balancing scenarios is into fixed time and infinite. Fixed time processes are only analyzed for an interval of time that is known in advance, and for all such results thus far either the number of rounds or the total expected number of arrivals at each server is a constant. In the infinite case, there is an arrival process and a deletion process that are both defined over an infinite time line. In this pape...
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 12 (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.
SIMLAB  A Simulation Environment for Storage Area Networks
 In Workshop on Parallel and Distributed Processing (PDP
, 2001
"... In this paper, we present a simulation environment for storage area networks called SIMLAB. SIMLAB is a part of the PRESTO project, which is a joint project of the Electrical Engineering Department and the Computer Science Department of the Paderborn University. The aim of the PRESTO project is to c ..."
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Cited by 5 (1 self)
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In this paper, we present a simulation environment for storage area networks called SIMLAB. SIMLAB is a part of the PRESTO project, which is a joint project of the Electrical Engineering Department and the Computer Science Department of the Paderborn University. The aim of the PRESTO project is to construct a scalable and resourceefficient storage network that can support the realtime delivery of data. SIMLAB has been implemented to aid the development and verification of distributed algorithms for this storage network. However, it has been designed in such a way that it can also be used for the simulation of many other types of networking problems. SIMLAB is based on C++ and common libraries and input/output formats, which ensures that SIMLAB can be used on many different platforms. We therefore expect SIMLAB to be useful also for other people working on similar problems. 1 Introduction In the last couple of years, a dramatic increase in the need of storing huge amounts of data can...
M.: Revisiting Randomized Parallel Load Balancing Algorithms
 In: Proc. 16th Colloquium on Structural Information and Communication Complexity (SIROCCO
, 2009
"... We deal with the well studied allocation problem of assigning n balls to n bins so that the maximum number of balls assigned to the same bin is minimized. We focus on randomized, constantround, distributed, asynchronous algorithms for this problem. Adler et al. [1] presented lower bounds and upper ..."
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We deal with the well studied allocation problem of assigning n balls to n bins so that the maximum number of balls assigned to the same bin is minimized. We focus on randomized, constantround, distributed, asynchronous algorithms for this problem. Adler et al. [1] presented lower bounds and upper bounds for this problem. A similar lower bound appears in Berenbrink et al. [3]. The lower bound is based on a topological assumption. Our first contribution is the observation that the topological assumption does not hold for two algorithms presented by Adler et al. [1]. We amend this situation by presenting direct proofs of the lower bound for these two algorithms. We present an algorithm in which a ball that was not allocated in the first round retries with a new choice in the second round. We present tight bounds on the maximum load obtained by our algorithm. The analysis is based on analyzing the expectation and transforming it to a bound with high probability using martingale tail inequalities. Finally, we present a 3round heuristic with a single synchronization point. We conducted experiments that demonstrate its advantage over parallel algorithms for 106 ≤ n ≤ 108 balls and bins. In fact, the obtained maximum load meets the best results for sequential algorithms.
M.: Parallel Randomized Load Balancing: A Lower Bound for a More General Model
 In: Proc. 36th Conference on Theory and Practice of Computer Science (SOFSEM
, 2010
"... We extend the lower bound of Adler et. al [1] and Berenbrink [3] for parallel randomized load balancing algorithms. The setting in these asynchronous and distributed algorithms is of n balls and n bins. The algorithms begin by each ball choosing d bins independently and uniformly at random. The ball ..."
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We extend the lower bound of Adler et. al [1] and Berenbrink [3] for parallel randomized load balancing algorithms. The setting in these asynchronous and distributed algorithms is of n balls and n bins. The algorithms begin by each ball choosing d bins independently and uniformly at random. The balls and bins communicate to determine the assignment of each ball to a bin. The goal is to minimize the maximum load, i.e., the number of balls that are assigned to the same bin. In [1, 3], a lower bound of Ω ( r logn / log log n) is proved if the communication is limited to r rounds. Three assumptions appear in the proofs in [1, 3]: the topological assumption, random choices of confused balls, and symmetry. We extend the proof of the lower bound so that it holds without these three assumptions. This lower bound applies to every parallel randomized load balancing algorithm we are aware of [1, 3, 8, 5].
Symmetric vs. Asymmetric MultipleChoice Algorithms
"... Multiplechoice allocation algorithms have been studied intensively over the last decade. These algorithms have several applications in the areas of load balancing, routing, resource allocation and hashing. The underlying idea is simple and can be explained best in the ballsandbins model: Instead ..."
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Multiplechoice allocation algorithms have been studied intensively over the last decade. These algorithms have several applications in the areas of load balancing, routing, resource allocation and hashing. The underlying idea is simple and can be explained best in the ballsandbins model: Instead of assigning balls (jobs, requests, or keys) simply at random to bins (machines, servers, or positions in a hash table), choose first a small set of bins at random, inspect these bins, and place the ball into one of the bins containing the smallest number of balls among them. The simple idea of first selecting a small set of alternatives at random and then making the final choice after careful inspection of these alternatives leads to great improvements against algorithms that place their decisions simply at random. We illustrate the power of this principle in terms of simple ballsandbins processes. In particular, we study recently presented algorithms that treat bins asymmetrically in order to obtain a better load balancing. We compare the behavior of these asymmetric schemes with symmetric schemes and prove that the asymmetric schemes achieve a better load balancing than their symmetric counterparts. 1
Randomized Allocation Processes (Extended Abstract)
, 1997
"... ) Artur Czumaj y Volker Stemann z Abstract We investigate various randomized processes allocating balls into bins that arise in applications in dynamic resource allocation and online load balancing. We consider the scenario when m balls arriving sequentially are to be allocated into n bins on ..."
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) Artur Czumaj y Volker Stemann z Abstract We investigate various randomized processes allocating balls into bins that arise in applications in dynamic resource allocation and online load balancing. We consider the scenario when m balls arriving sequentially are to be allocated into n bins online and without using a global controller. Traditionally, the main aim of allocation processes is to place the balls into bins to minimize the maximum load in bins. However, in many applications it is equally important to minimize the number of trails performed by the balls (the allocation time). We study adaptive allocation schemes that achieve optimal tradeoffs between the maximum load, the maximum allocation time, and the average allocation time. We investigate allocation processes that may reallocate the balls. We provide a tight analysis of the maximum load of processes that during placing a new ball may reassign the balls in up to d randomly chosen bins. We study infinite processe...