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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.
Balls into Bins via Local Search
, 2012
"... We propose a natural process for allocating n balls into n bins that are organized as the vertices of an undirected graph G. Each ball first chooses a vertex u in G uniformly at random. Then the ball performs a local search in G starting from u until it reaches a vertex with local minimum load, wher ..."
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Cited by 4 (1 self)
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We propose a natural process for allocating n balls into n bins that are organized as the vertices of an undirected graph G. Each ball first chooses a vertex u in G uniformly at random. Then the ball performs a local search in G starting from u until it reaches a vertex with local minimum load, where the ball is finally placed on. In our main result, we prove thatthisprocessyieldsamaximumloadofonlyΘ(loglogn)onexpandergraphs. Inaddition, ( ( logn we show that for ddimensional grids the maximum load is Θ loglogn) 1 d+1. Finally, for almost regular graphs with minimum degree Ω(logn), we prove that the maximum load is constantandalsorevealafundamentaldifferencebetweenrandomandarbitrarytiebreaking rules. 1
DISS. ETH NO. 19459 Synchronization and Symmetry Breaking in Distributed Systems
, 2010
"... accepted on the recommendation of ..."
Distributed Computing manuscript No. (will be inserted by the editor) Tight Bounds for Parallel Randomized Load Balancing
"... Abstract Given a distributed system of n balls and n bins, how evenly can we distribute the balls to the bins, minimizing communication? The fastest nonadaptive and symmetric algorithm achieving a constant maximum bin load requires Θ(log log n) rounds, and any such algorithm running for r ∈ O(1) r ..."
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Abstract Given a distributed system of n balls and n bins, how evenly can we distribute the balls to the bins, minimizing communication? The fastest nonadaptive and symmetric algorithm achieving a constant maximum bin load requires Θ(log log n) rounds, and any such algorithm running for r ∈ O(1) rounds incurs a bin load of Ω((log n / log log n)1/r). In this work, we explore the fundamental limits of the general problem. We present a simple adaptive symmetric algorithm that achieves a bin load of 2 in log ∗ n + O(1) communication rounds using O(n) messages in total. Our main result, however, is a matching lower bound of (1 − o(1)) log ∗ n on the time complexity of symmetric algorithms that guarantee small bin loads. 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 need not be 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. An extended abstract of preliminary work appeared at STOC 2011 [24] and the corresponding article has been published on arxiv [23].
Pursuing the Giant in Random Graph Processes
, 2013
"... We study the evolution of random graph processes that are based on the paradigm of the power of multiple choices. The processes we consider begin with an empty graph on n vertices. In each subsequent step a set with a specific number ℓ ≥ 2 of random vertices is presented, and we may select any edge ..."
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We study the evolution of random graph processes that are based on the paradigm of the power of multiple choices. The processes we consider begin with an empty graph on n vertices. In each subsequent step a set with a specific number ℓ ≥ 2 of random vertices is presented, and we may select any edge among them to be included in the graph. For example, if ℓ = 2 this corresponds to the classical ErdősRényi (ER) process. A striking characteristic of the ER process is the phase transition with respect to the distribution of its component sizes. This distribution undergoes a drastic change when the number of edges is around n/2; at this point the socalled giant component emerges, which contains a linear fraction of the vertices. In this paper we address the componentsize distribution of a general family rules. We determine the typical size of the giant component shortly after the phase transition in all these processes and provide bounds for the size distribution of small components. In particular, it has been conjectured by various authors that these processes have many similarities with the ER process, for example that the giant component grows with a constant “rate”. Our results confirm this conjecture. On the technical side, we develop a novel method for the analysis of the component size distribution based on partial differential equations (PDEs). We develop a novel analytic framework that allows us to study the solutions of a fairly general class of quasilinear PDEs, the socalled family of Smoluchowski’s coagulation equations, that have several farreaching applications in the study of large systems consisting of interacting particles. Finally, our family of rules allows us to “approximate ” formally any general size rule by a sequence of appropriately defined bounded size rules, where the given sizebound increases gradually. Thus, our results open an avenue for the future research on general random graph processes. 1 1
Multidimensional Balanced Allocation for Multiple Choice & (1 + β) Processes
, 2011
"... copyrighted is accepted for publication. It has been issued as a Research Report for the early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requ ..."
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copyrighted is accepted for publication. It has been issued as a Research Report for the early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g., payment of royalties). Copies may be requested from IBM T.J. Watson Research Center, Publications,
Balancing indivisible realvalued loads in arbitrary networks
, 2014
"... In parallel computing, a problem is divided into a set of smaller tasks that are distributed across multiple processing elements. Balancing the load of the processing elements is key to achieving good performance and scalability. If the computational costs of the individual tasks vary over time in a ..."
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In parallel computing, a problem is divided into a set of smaller tasks that are distributed across multiple processing elements. Balancing the load of the processing elements is key to achieving good performance and scalability. If the computational costs of the individual tasks vary over time in an unpredictable way, dynamic load balancing aims at migrating them between processing elements so as to maintain load balance. During dynamic load balancing, the tasks amount to indivisible work packets with a realvalued cost. For this case of indivisible, realvalued loads, we analyze the balancing circuit model, a local dynamic loadbalancing scheme that does not require global communication. We extend previous analyses to the present case and provide a probabilistic bound for the achievable load balance. Based on an analogy with the offline ballsintobins problem, we further propose a novel algorithm for dynamic balancing of indivisible, realvalued loads. We benchmark the proposed algorithm in numerical experiments and compare it with the classical greedy algorithm, both in terms of solution quality and communication cost. We find that the increased communication cost of the proposed algorithm is compensated by a higher solution quality, leading on average to about an order of magnitude gain in overall performance.
Balanced offline allocation of weighted balls into bins
"... We propose a sortingbased greedy algorithm called SortedGreedy[m] for approximately solving the offline version of the dchoice weighted ballsintobins problem where the number of choices for each ball is equal to the number of bins. We assume the ball weights to be nonnegative. We compare the pe ..."
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We propose a sortingbased greedy algorithm called SortedGreedy[m] for approximately solving the offline version of the dchoice weighted ballsintobins problem where the number of choices for each ball is equal to the number of bins. We assume the ball weights to be nonnegative. We compare the performance of the sortingbased algorithm with a näıve algorithm called Greedy[m]. We show that by sorting the input data according to the weights we are able to achieve an order of magnitude smaller gap (the weight difference between the heaviest and the lightest bin) for small problems ( ≤ 4000 balls), and at least two orders of magnitude smaller gap for larger problems. In practice, SortedGreedy[m] runs almost as fast as Greedy[m]. This makes sortingbased algorithms favorable for solving offline weighted ballsintobins problems.