This paper presents a distributed O(1)-approximation algorithm, with expected- @math running time, in the @math model for the metric facility location problem on a size- @math clique network. Though metric facility location has been considered by a number of researchers in low-diameter settings, this is the first sub-logarithmic-round algorithm for the problem that yields an O(1)-approximation in the setting of non-uniform facility opening costs. In order to obtain this result, our paper makes three main technical contributions. First, we show a new lower bound for metric facility location, extending the lower bound of B a (ICALP 2005) that applies only to the special case of uniform facility opening costs. Next, we demonstrate a reduction of the distributed metric facility location problem to the problem of computing an O(1)-ruling set of an appropriate spanning subgraph. Finally, we present a sub-logarithmic-round (in expectation) algorithm for computing a 2-ruling set in a spanning subgraph of a clique. Our algorithm accomplishes this by using a combination of randomized and deterministic sparsification.
Several more recent papers have continued the development of super-fast'' algorithms in low-diameter settings. In STOC 2011, Lenzen and Wattenhofer @cite_16 derived tight bounds on parallel load balancing and their result has applications in how information can be quickly disseminated in a clique (in the @math model). In PODC 2011, Patt-Shamir and Teplitsky @cite_20 presented on @math randomized algorithm for the problem. Third, Lenzen @cite_2 showed that randomization is not necessary for solving problems such as distributed sorting efficiently. Lenzen presented , constant-round algorithms for a routing problem and for the distributed sorting problem considered in @cite_20 . Constant-round algorithms for sophisticated problems, of the kind described by Lenzen @cite_2 , highlight the difficulty of showing non-trivial lower bounds in the @math model for clique networks. For example, it has been proved that computing an MST in general requires @math rounds for diameter- @math graphs @cite_3 , but no non-trivial lower bounds are known for diameter- @math or clique (diameter- @math ) networks.
@cite_16 @cite_3 @cite_20 @cite_2
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We explore the fundamental limits of distributed balls-into-bins algorithms, i.e., algorithms where balls act in parallel, as separate agents. This problem was introduced by , who showed that non-adaptive and symmetric algorithms cannot reliably perform better than a maximum bin load of Theta(log log n log log log n) 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. This paper considers the problem of distributively constructing a minimum-weight spanning tree (MST) for graphs of constant diameter in the bounded-messages model, where each message can contain at most B bits for some parameter B. It is shown that the number of communication rounds necessary to compute an MST for graphs of diameter 4 or 3 can be as high as ( ( [3]n B ) ) and ( ( [4]n B ) ), respectively. The asymptotic lower bounds hold for randomized algorithms as well. On the other hand, we observe that O(log n) communication rounds always suffice to compute an MST deterministically for graphs with diameter 2, when B = O(log n). These results complement a previously known lower bound of ( ( [2]n B) ) for graphs of diameter Ω(log n). We consider the model of fully connected networks, where in each round each node can send an O(log n)-bit message to each other node (this is the CONGEST model with diameter 1). It is known that in this model, min-weight spanning trees can be found in O(log log n) rounds. In this paper we show that distributed sorting, where each node has at most n items, can be done in time O(log log n) as well. It is also shown that selection can be done in O(1) time. (Using a concurrent result by Lenzen and Wattenhofer, the complexity of sorting is further reduced to constant.) Our algorithms are randomized, and the stated complexity bounds hold with high probability. Consider a clique of n nodes, where in each synchronous round each pair of nodes can exchange O(log n) bits. We provide deterministic constant-time solutions for two problems in this model. The first is a routing problem where each node is source and destination of n messages of size O(log n). The second is a sorting problem where each node i is given n keys of size O(log n) and needs to receive the ith batch of n keys according to the global order of the keys. The latter result also implies deterministic constant-round solutions for related problems such as selection or determining modes.