Randomized Parallel Selection

We show that selection on an input of size N can be performed on a P-node hypercube (P = N/(log N)) in time O(n/P) with high probability, provided each node can process all the incident edges in one unit of time (this model is called the parallel model and has been assumed by previous researchers (e.g.,[17])). This result is important in view of a lower bound of Plaxton that implies selection takes Ω((N/P)loglog P+log P) time on a P-node hypercube if each node can process only one edge at a time (this model is referred to as the sequential model)

Constant Queue Route on a Mesh

Packet routing is an important problem in parallel computation since a single step of inter-processor communication can be thought of as a packet routing task. In this paper we present an optimal algorithm for packet routing on a mesh-connected computer. Two important criteria for judging a routing algorithm will be 1) its run time, i.e., the number of parallel steps it takes for the last packet to reach its destination, and 2) its queue size, i.e., the maximum number of packets that any node will have to store at any time during routing. We present a 2n - 2 step routing algorithm for an n x n mesh that requires a queue size of only 58. The previous best known result is a routing algorithm with the same time bound but with a queue size of 672. The time bound of 2n - 2 is optimal. A queue size of 672 is rather large for practical use. We believe that the queue size of our algorithm is practical. The improvement in the queue size is possible due to (from among other things) a new 3s + o(s) sorting algorithm for an s x s mesh

Singletons for Simpletons: Revisiting Windowed Backoff with Chernoff Bounds

Backoff algorithms are used in many distributed systems where multiple devices contend for a shared resource. For the classic balls-into-bins problem, the number of singletons - those bins with a single ball - is important to the analysis of several backoff algorithms; however, existing analyses employ advanced probabilistic tools to obtain concentration bounds. Here, we show that standard Chernoff bounds can be used instead, and the simplicity of this approach is illustrated by re-analyzing some well-known backoff algorithms

Randomized Routing on Fat-Trees

Fat-trees are a class of routing networks for hardware-efficient parallel computation. This paper presents a randomized algorithm for routing messages on a fat-tree. The quality of the algorithm is measured in terms of the load factor of a set of messages to be routed, which is a lower bound on the time required to deliver the messages. We show that if a set of messages has load factor lambda on a fat-tree with n processors, the number of delivery cycles (routing attempts) that the algorithm requires is O(lambda + lg n lg lg n) with probability 1-O(1/n). The best previous bound was O(lambda lg n) for the offline problem in which the set of messages is known in advance. In the context of a VLSI model that equates hardware cost with physical volume, the routing algorithm can be used to demonstrate that fat-trees are universal routing networks. Specifically, we prove that any routing network can be efficiently simulated by a fat-tree of comparable hardware cost

Performance of hypercube routing schemes with or without buffering

Includes bibliographical references (p. 34-35).Supported by the NSF. NSF-DDM-8903385 Supported by the ARO. DAAL03-92-G-0115by Emmanouel A. Varvarigos and Dimitri P. Bertsekas

Hop-Constrained Oblivious Routing

We prove the existence of an oblivious routing scheme that is $\mathrm{poly}(\log n)$-competitive in terms of $(congestion + dilation)$, thus resolving a well-known question in oblivious routing. Concretely, consider an undirected network and a set of packets each with its own source and destination. The objective is to choose a path for each packet, from its source to its destination, so as to minimize $(congestion + dilation)$, defined as follows: The dilation is the maximum path hop-length, and the congestion is the maximum number of paths that include any single edge. The routing scheme obliviously and randomly selects a path for each packet independent of (the existence of) the other packets. Despite this obliviousness, the selected paths have $(congestion + dilation)$ within a $\mathrm{poly}(\log n)$ factor of the best possible value. More precisely, for any integer hop-bound $h$, this oblivious routing scheme selects paths of length at most $h \cdot \mathrm{poly}(\log n)$ and is $\mathrm{poly}(\log n)$-competitive in terms of $congestion$ in comparison to the best possible $congestion$ achievable via paths of length at most $h$ hops. These paths can be sampled in polynomial time. This result can be viewed as an analogue of the celebrated oblivious routing results of R\"{a}cke [FOCS 2002, STOC 2008], which are $O(\log n)$-competitive in terms of $congestion$, but are not competitive in terms of $dilation$

Efficient routing schemes for multiple broadcasts in hypercubes

"February 1990/Revised June 1990."--Cover. Cover title.Includes bibliographical references (p. 36-37).Research supported by the NSF. ECS-8552419 Research supported by Bellcore, Inc. and Du Pont. Research supported by the ARO. DAAL03-86-K-0171 Research supported by a fellowship from the Vinton Hayes Fund.George D. Stamoulis and John N. Tsitsiklis

Coping with dynamic membership, selfishness, and incomplete information: applications of probabilistic analysis and game theory

textThe emergence of large scale distributed computing networks has given increased prominence to a number of algorithmic concerns, including the need to handle dynamic membership, selfishness, and incomplete information. In this document, we outline our explorations into these algorithmic issues. We first present our results on the analysis of a graph-based coupon collecvi tor process related to load balancing for networks with dynamic membership. In addition to extending the study of the coupon collector process, our results imply load balancing properties of certain distributed hash tables. Second, we detail our results on worst case payoffs when playing buyersupplier games, against many selfish, collaborating opponents. We study optimization over the set of core vectors. We show both positive and negative results on optimizing over the cores of such games. Furthermore, we introduce and study the concept of focus point price, which answers the question: If we are constrained to play in equilibrium, how much can we lose by playing the wrong equilibrium? Finally, we present our analysis of a revenue management problem with incomplete information, the online weighted transversal matroid matching problem. In specific, we present an algorithm that delivers expected revenue within a constant of optimal in the online setting. Our results use a novel algorithm to generalize several results known for special cases of transversal matroids.Computer Science