We study the problem of power-efficient routing for multihop wireless ad hoc sensor networks. The guiding insight of our work is that unlike an ad hoc wireless network, a wireless ad hoc sensor network does not require full connectivity among the nodes. As long as the sensing region is well covered by connected nodes, the network can perform its task. We consider two kinds of geometric random graphs as base interconnection structures: unit disk graphs @math and @math -nearest-neighbor graphs @math built on points generated by a Poisson point process of density @math in @math . We provide subgraph constructions for these two models @math and @math and show that there are values @math and @math above which these constructions have the following good properties: (i) they are sparse; (ii) they are power-efficient in the sense that the graph distance is no more than a constant times the Euclidean distance between any pair of points; (iii) they cover the space well; (iv) the subgraphs can be set up easily using local information at each node. We also describe a simple local algorithm for routing packets on these subgraphs. Our constructions also give new upper bounds for the critical values of the parameters @math and @math for the models @math and @math .
The @math model was introduced by H "aggstr "om and Meester @cite_14 . They showed that there was a finite critical value, @math for all @math such that an infinite cluster exists in this model. They proved that the infinite cluster was unique and that there was a value @math such that @math for all @math . Teng and Yao gave an upper bound of 213 for @math @cite_15 .
@cite_15 @cite_14
2047805348 2261587243 1914749871 2000931246
We prove the first nontrivial (superlinear) lower bound in the noisy broadcast model, defined by El Gamal in [Open problems presented at the @math workshop on Specific Problems in Communication and Computation sponsored by Bell Communication Research, in Open Problems in Communication and Computation, T. M. Cover and B. Gopinath, eds., Springer-Verlag, New York, 1987, pp. 60-62]. In this model there are @math processors @math , each of which is initially given a private input bit @math . The goal is for @math to learn the value of @math , for some specified function @math , using a series of noisy broadcasts. At each step a designated processor broadcasts one bit to all of the other processors, and the bit received by each processor is flipped with fixed probability (independently for each recipient). In 1988, Gallager [IEEE Trans. Inform. Theory, 34 (1988), pp. 176-180] gave a noise-resistant protocol that allows @math to learn the entire input with constant probability in @math broadcasts. We prove that Gallager's protocol is optimal, up to a constant factor. Our lower bound follows by reduction from a lower bound for generalized noisy decision trees, a new model which may be of independent interest. For this new model we show a lower bound of @math on the depth of a tree that learns the entire input. While the above lower bound is for an @math -bit function, we also show an @math lower bound for the number of broadcasts required to compute certain explicit boolean-valued functions, when the correct output must be attained with probability at least @math for a constant parameter @math (this bound applies to all threshold functions as well as any other boolean-valued function with linear sensitivity). This bound also follows by reduction from a lower bound of @math on the depth of generalized noisy decision trees that compute the same functions with the same error. We also show a (nontrivial) @math lower bound on the depth of generalized noisy decision trees that compute such functions with small constant error. Finally, we show the first protocol in the noisy broadcast model that computes the Hamming weight of the input using a linear number of broadcasts. In this paper, we are interested in the loop cluster model on @math for @math . It is a long range model with two parameters @math and @math , where the non-negative parameter @math measures the amount of loops, and @math plays the role of killing on vertices penalizing ( @math ) or favoring ( @math large balls in the infinite cluster are finally very regular in the sense of Sapozhnikov2014 , which implies that large balls are finally very good in the sense of BarlowMR2094438 . By BarlowMR2094438 and BarlowHamblyMR2471657 , we have Harnack's inequality and Gaussian type estimate for simple random walks on the infinite cluster for all @math . We study a random graph model named the "block model" in statistics and the "planted partition model" in theoretical computer science. In its simplest form, this is a random graph with two equal-sized clusters, with a between-class edge probability of @math and a within-class edge probability of @math . A striking conjecture of Decelle, Krzkala, Moore and Zdeborov 'a based on deep, non-rigorous ideas from statistical physics, gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if @math and @math , @math and @math then conjectured that it is possible to efficiently cluster in a way correlated with the true partition if @math and impossible if @math for some sufficiently large @math . In a previous work, we proved that indeed it is information theoretically impossible to to cluster if @math . A different independent proof of the same result was recently obtained by Laurent Massoulie. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253