We study greedy routing over @math nodes placed in a ring, with the between two nodes defined to be the clockwise or the absolute distance between them along the ring. Such graphs arise in the context of modeling social networks and in routing networks for peer-to-peer systems. We construct the first network over @math nodes in which greedy routing takes @math hops in the worst-case, with @math out-going links per node. Our result has the first asymptotically optimal greedy routing complexity. Previous constructions required @math hops.
Viceroy @cite_14 is a butterfly network which routes in @math hops in expectation with @math links per node. Mariposa (see reference @cite_23 or @cite_21 ) improves upon Viceroy by providing routes of length @math in the worst-case, with @math out-going links per node. Viceroy and Mariposa are different from other randomized networks in terms of their design philosophy. The topology borrows elements of the geometric embedding of the butterfly in a circle from Viceroy @cite_14 and from @cite_21 , while extending them for greedy routing.
@cite_14 @cite_21 @cite_23
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Given @math wireless transceivers located in a plane, a fundamental problem in wireless communications is to construct a strongly connected digraph on them such that the constituent links can be scheduled in fewest possible time slots, assuming the SINR model of interference. In this paper, we provide an algorithm that connects an arbitrary point set in @math slots, improving on the previous best bound of @math due to Moscibroda. This is complemented with a super-constant lower bound on our approach to connectivity. An important feature is that the algorithms allow for bi-directional (half-duplex) communication. One implication of this result is an improved bound of @math on the worst-case capacity of wireless networks, matching the best bound known for the extensively studied average-case. We explore the utility of oblivious power assignments, and show that essentially all such assignments result in a worst case bound of @math slots for connectivity. This rules out a recent claim of a @math bound using oblivious power. On the other hand, using our result we show that @math slots suffice, where @math is the ratio between the largest and the smallest links in a minimum spanning tree of the points. Our results extend to the related problem of minimum latency aggregation scheduling, where we show that aggregation scheduling with @math latency is possible, improving upon the previous best known latency of @math . We also initiate the study of network design problems in the SINR model beyond strong connectivity, obtaining similar bounds for biconnected and @math -edge connected structures. In this paper we study randomized algorithms for circuit switching on multistage networks related to the butterfly. We devise algorithms that route messages by constructing circuits (or paths) for the messages with small congestion, dilation, and setup time. Our algorithms are based on the idea of having each message choose a route from two possibilities, a technique that has previously proven successful in simpler load balancing settings. As an application of our techniques, we propose a novel design for a data server. We consider the task of topology discovery of sparse random graphs using end-to-end random measurements (e.g., delay) between a subset of nodes, referred to as the participants. The rest of the nodes are hidden, and do not provide any information for topology discovery. We consider topology discovery under two routing models: (a) the participants exchange messages along the shortest paths and obtain end-to-end measurements, and (b) additionally, the participants exchange messages along the second shortest path. For scenario (a), our proposed algorithm results in a sub-linear edit-distance guarantee using a sub-linear number of uniformly selected participants. For scenario (b), we obtain a much stronger result, and show that we can achieve consistent reconstruction when a sub-linear number of uniformly selected nodes participate. This implies that accurate discovery of sparse random graphs is tractable using an extremely small number of participants. We finally obtain a lower bound on the number of participants required by any algorithm to reconstruct the original random graph up to a given edit distance. We also demonstrate that while consistent discovery is tractable for sparse random graphs using a small number of participants, in general, there are graphs which cannot be discovered by any algorithm even with a significant number of participants, and with the availability of end-to-end information along all the paths between the participants. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013 In classical routing strategies for multihop mobile wireless networks packets are routed on a pre-defined route usually obtained by a shortest path routing protocol. In opportunistic routing schemes, for each packet and each hop, the next relay is found by dynamically selecting the node that captures the packet transmission and which is the nearest to the destination. Such a scheme allows each packet to take advantage of the local pattern of transmissions and fadings at any slot and at any hop. The aim of this paper is to quantify and optimize the potential performance gains of such opportunistic routing strategies compared with classical routing schemes. The analysis is conducted under the following lower layer assumptions: the Medium Access (MAC) layer is a spatial version of Aloha which has been shown to scale well for large multihop networks; the capture of a packet by some receiver is determined by the Signal over Interference and Noise Ratio (SINR) experienced by the receiver. The paper contains a detailed simulation study which shows that such time-space opportunistic schemes very significantly outperform classical routing schemes. It also contains a mathematical study where we show how to optimally tune the MAC parameters so as to minimize the average number of time slots required to carry a typical packet from origin to destination on long paths. We show that this optimization is independent of network density.