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.,...

This paper deals with balls and bins processes related to randomized load balancing, dynamic resource allocation, and hashing. Suppose ¡ balls have to be assigned to ¡ bins, where each ball has to be placed without knowledge about the distribution of previously placed balls. The goal is to achieve an allocation that is as even as possible so that no bin gets much more balls than the average. A well known and good solution for this problem is to choose ¢ possible locations for each ball at random, to look into each of these bins, and to place the ball into the least full among these bins. This class of algorithms has been investigated intensively in the past, but almost all previous analyses assume that the ¢ locations for each ball are chosen uniformly and independently at random from the set of all bins. We investigate whether a non-uniform and possibly de-pendent choice of the ¢

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 ...

High performance applications involving large data sets require the efficient and flexible use of multiple disks. In an external memory machine with D parallel, independent disks, only one block can be accessed on each disk in one I/O step. This restriction leads to a load balancing problem that is perhaps the main inhibitor for the efficient adaptation of single-disk external memory algorithms to multiple disks. We solve this problem for arbitrary access patterns by randomly mapping blocks of a logical address space to the disks. We show that a shared buffer of O(D) blocks suffices to support efficient writing. The analysis uses the properties of negative association to handle dependencies between the random variables involved. This approach might be of independent interest for probabilistic analysis in general. If two randomly allocated copies of each block exist, N arbitrary blocks can be read within dN=De + 1 I/O steps with high probability. The redundancy can be further reduced from 2 to 1 + 1=r for any integer r without a big impact on reading efficiency. From the point of view of external memory models, these results rehabilitate Aggarwal and Vitter's "single-disk multi-head" model [1] that allows access to D arbitrary blocks in each I/O step. This powerful model can be emulated on the physically more realistic independent disk model [2] with small constant overhead factors. Parallel disk external memory algorithms can therefore be developed in the multi-head model first. The emulation result can then be applied directly or further refinements can be added.

We survey routing problems on fixed-connection networks. We consider many aspects of the routing problem and provide known theoretical results for various communication models. We focus on (partial) permutation, k-relation routing, routing to random destinations, dynamic routing, isotonic routing, fault tolerant routing, and related sorting results. We also provide a list of unsolved problems and numerous references.

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...

We investigate a linear time greedy algorithm for the following load balancing problem: Assign m balls to n bins such that the maximum occupancy is minimized. Each ball can be placed into one of two randomly choosen bins. This problem is closely related to the problem of orienting the edges of an undirected graph to obtain a directed graph with minimum in-degree. Using differential equation methods, we derive thresholds for the solution quality achieved by our algorithm. Since these thresholds coincide with lower bounds for the achievable solution quality, this proves the optimality of our algorithm (as n → ∞, in a probabilistic sense) and establishes the thresholds for k-orientability of random graphs. This proves an assertion of Karp and Saks.

The study of hashing is closely related to the analysis of balls and bins. Azar et. al. [1] showed that instead of using a single hash function if we randomly hash a ball into two bins and place it in the smaller of the two, then this dramatically lowers the maximum load on bins. This leads to the concept of two-way hashing where the largest bucket contains O(log log n) balls with high probability. The hash look up will now search in both the buckets an item hashes to. Since an item may be placed in one of two buckets, we could potentially move an item after it has been initially placed to reduce maximum load. Using this fact, we present a simple, practical hashing scheme that maintains a maximum load of 2, with high probability, while achieving high memory utilization. In fact, with n buckets, even if the space for two items are pre-allocated per bucket, as may be desirable in hardware implementations, more than n items can be stored giving a high memory utilization. Assuming truly random hash functions, we prove the following properties for our hashing scheme. • Each lookup takes two random memory accesses, and reads at most two items per access. • Each insert takes O(log n) time and up to log log n+ O(1) moves, with high probability, and constant time in expectation. • Maintains 83.75 % memory utilization, without requiring dynamic allocation during inserts. We also analyze the trade-off between the number of moves performed during inserts and the maximum load on a bucket. By performing at most h moves, we can maintain a maximum load of O(hlogl((~og~og:n/h)). So, even by performing one move, we achieve a better bound than by performing no moves at all.

) Artur Czumaj 1 , Friedhelm Meyer auf der Heide 2 , and Volker Stemann 1 1 Heinz Nixdorf Institute, University of Paderborn, D-33095 Paderborn, Germany 2 Heinz Nixdorf Institute and Department of Computer Science, University of Paderborn, D-33095 Paderborn, Germany Abstract. We consider the problem of simulating a PRAM on a distributed memory machine (DMM). Our main result is a randomized algorithm that simulates each step of an n-processor CRCW PRAM on an n-processor DMM with O(log log log n log n) delay, with high probability. This is an exponential improvement on all previously known simulations. It can be extended to a simulation of an (n log log log n log n)- processor EREW PRAM on an n-processor DMM with optimal delay O(log log log n log n), with high probability. Finally a lower bound of \Omega (log log log n=log log log log n) expected time is proved for a large class of randomized simulations that includes all known simulations. 1 Introduction Para...

We explore the fundamental limits of distributed balls-intobins algorithms, i.e., algorithms where balls act in parallel, as separate agents. This problem was introduced by Adler et al., who showed that non-adaptive 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.