We study the wireless scheduling problem in the SINR model. More specifically, given a set of @math links, each a sender-receiver pair, we wish to partition (or ) the links into the minimum number of slots, each satisfying interference constraints allowing simultaneous transmission. In the basic problem, all senders transmit with the same uniform power. We give a distributed @math -approximation algorithm for the scheduling problem, matching the best ratio known for centralized algorithms. It holds in arbitrary metric space and for every length-monotone and sublinear power assignment. It is based on an algorithm of Kesselheim and V "ocking, whose analysis we improve by a logarithmic factor. We show that every distributed algorithm uses @math slots to schedule certain instances that require only two slots, which implies that the best possible absolute performance guarantee is logarithmic.
In @cite_11 , the authors introduce a versatile measure, the maximum average affectance @math , defined by @math The authors then show two results. On the one hand, they show that @math where @math . On the other hand, they present a natural algorithm (we use the same algorithm in this work) which schedules all links in @math slots, thus achieving a @math approximation. We can show (Appendix ) that both of these bounds are tight. Thus it is not possible to obtain improved approximation using the measure @math .
@cite_11
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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 We present and analyze simple distributed contention resolution protocols for wireless networks. In our setting, one is given n pairs of senders and receivers located in a metric space. Each sender wants to transmit a signal to its receiver at a prespecified power level, e. g., all senders use the same, uniform power level as it is typically implemented in practice. Our analysis is based on the physical model in which the success of a transmission depends on the Signal-to-Interference-plus-Noise-Ratio (SINR). The objective is to minimize the number of time slots until all signals are successfully transmitted. Our main technical contribution is the introduction of a measure called maximum average affectance enabling us to analyze random contention-resolution algorithms in which each packet is transmitted in each step with a fixed probability depending on the maximum average affectance. We prove that the schedule generated this way is only an O(log2 n) factor longer than the optimal one, provided that the prespecified power levels satisfy natural monontonicity properties. By modifying the algorithm, senders need not to know the maximum average affectance in advance but only static information about the network. In addition, we extend our approach to multi-hop communication achieving the same appoximation factor. 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. Let @math be a tree space (or tree network) represented by a weighted tree with @math vertices, and @math be a set of @math stochastic points in @math , each of which has a fixed location with an independent existence probability. We investigate two fundamental problems under such a stochastic setting, the closest-pair problem and the nearest-neighbor search. For the former, we study the computation of the @math -threshold probability and the expectation of the closest-pair distance of a realization of @math . We propose the first algorithm to compute the @math -threshold probability in @math time for any given threshold @math , which immediately results in an @math -time algorithm for computing the expected closest-pair distance. Based on this, we further show that one can compute a @math -approximation for the expected closest-pair distance in @math time, by arguing that the expected closest-pair distance can be approximated via @math threshold probability queries. For the latter, we study the @math most-likely nearest-neighbor search ( @math -LNN) via a notion called @math most-likely Voronoi Diagram ( @math -LVD). We show that the size of the @math -LVD @math of @math on @math is bounded by @math if the existence probabilities of the points in @math are constant-far from 0. Furthermore, we establish an @math average-case upper bound for the size of @math , by regarding the existence probabilities as i.i.d. random variables drawn from some fixed distribution. Our results imply the existence of an LVD data structure which answers @math -LNN queries in @math time using average-case @math space, and worst-case @math space if the existence probabilities are constant-far from 0. Finally, we also give an @math -time algorithm to construct the LVD data structure.