We consider the local broadcasting problem in the SINR model, which is a basic primitive for gathering initial information among @math wireless nodes. Assuming that nodes can measure received power, we achieve an essentially optimal constant approximate algorithm (with a @math additive term). This improves upon the previous best @math -approximate algorithm. Without power measurement, our algorithm achieves @math -approximation, matching the previous best result, but with a simpler approach that works under harsher conditions, such as arbitrary node failures. We give complementary lower bounds under reasonable assumptions.
The bounds for unknown competition were improved in @cite_17 to @math , optimizing the algorithm of @cite_1 . Additionally, @math -approximate deterministic algorithms were given for a synchronized model where a carrier-sense primitive was assumed to be available. Finally, an @math -approximate randomized algorithm without a carrier-sense primitive was very recently proposed in @cite_19 . A lower bound of @math was also given.
@cite_19 @cite_1 @cite_17
2595267259 2000931246 2168012204 2123485784
In this paper we improve the deterministic complexity of two fundamental communication primitives in the classical model of ad hoc radio networks with unknown topology: broadcasting and wake-up. We consider an unknown radio network, in which all nodes have no prior knowledge about network topology, and know only the size of the network @math , the maximum in-degree of any node @math , and the eccentricity of the network @math . For such networks, we first give an algorithm for wake-up, based on the existence of small universal synchronizers. This algorithm runs in @math time, the fastest known in both directed and undirected networks, improving over the previous best @math -time result across all ranges of parameters, but particularly when maximum in-degree is small. Next, we introduce a new combinatorial framework of block synchronizers and prove the existence of such objects of low size. Using this framework, we design a new deterministic ... 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 study an online unit-job scheduling problem arising in buffer management. Each job is specified by its release time, deadline, and a nonnegative weight. Due to overloading conditions, some jobs have to be dropped. The goal is to maximize the total weight of scheduled jobs. We present several competitive online algorithms for various versions of unit-job scheduling, as well as some lower bounds on the competitive ratios. We first give a randomized algorithm RMix with competitive ratio of e (e-1) 1.582. This is the first algorithm for this problem with competitive ratio smaller than 2. Then we consider s-bounded instances, where the span of each job (deadline minus release time) is at most s. We give a 1.25-competitive randomized algorithm for 2-bounded instances, matching the known lower bound. We also give a deterministic algorithm Edf"@a, whose competitive ratio on s-bounded instances is 2-2 s+o(1 s). For 3-bounded instances its ratio is @f 1.618, matching the known lower bound. In s-uniform instances, the span of each job is exactly s. We show that no randomized algorithm can be better than 1.25-competitive on s-uniform instances, if the span s is unbounded. For s=2, our proof gives a lower bound of 4-22 1.172. Also, in the 2-uniform case, we prove a lower bound of 2 1.414 for deterministic memoryless algorithms, matching a known upper bound. Finally, we investigate the multiprocessor case and give a 1 (1-(mm+1)^m)-competitive algorithm for m processors. We also show improved lower bounds for the general and s-uniform cases. The problem of finding high-dimensional approximate nearest neighbors is considered when the data is generated by some known probabilistic model. A large natural class of algorithms (bucketing codes) is investigated, Bucketing information is defined, and is proven to bound the performance of all bucketing codes. The bucketing information bound is asymptotically attained by some randomly constructed bucketing codes. The example of n Bernoulli(1 2) very long (length d → ∞) sequences of bits is singled out. It is assumed that n - 2m sequences are completely independent, while the remaining 2m sequences are composed of m dependent pairs. The interdependence within each pair is that their bits agree with probability 1 2 0. A specific 2-D inequality (proven in another paper) implies that the exponent 1 p cannot be lowered. Moreover, if one sequence out of each pair belongs to a known set of n(2p-1)2 sequences, pairing can be done using order n1+∈ comparisons!