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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 ..."
Abstract

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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.
DISS. ETH NO. 19459 Synchronization and Symmetry Breaking in Distributed Systems
, 2010
"... accepted on the recommendation of ..."
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 ..."
Abstract
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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