Abstract. In this paper we present a new heuristic called Adaptive Broadcast Consumption (ABC for short) for the Minimum-Energy Broadcast Routing (MEBR) problem. We first investigate the problern trying to understand which are the main properties not taken into account by the dassie and well-studied MST and BIP heuristics, then we propose a new algorithm proving that it computes the MEBR with an approximation ratio less than or equal to MST, for which we prove an approximation ratio of at most 12.15 instead of the well-known 12 [10]. Finally we present experimental results supporting our intuitive ideas, comparing ABC with other heuristics presented in the literature and showing its good performance on random instances even compared to the optimum.

Given any multiset F of points in the Euclidean plane and a set R of robots such that |R| = |F|, the Arbitrary Pattern Formation (APF) problem asks for a distributed algorithm that moves robots so as to reach a configuration similar to F. Similarity means that robots must be disposed as F regardless of translations, rotations, reflections, uniform scalings. Initially, each robot occupies a distinct position. When active, a robot operates in standard Look-ComputeMove cycles. Robots are asynchronous, oblivious, anonymous, silent and execute the same distributed algorithm. So far, the problem has been mainly addressed by assuming chirality, that is robots share a common left-right orientation. We are interested in removing such a restriction.While working on the subject, we faced several issues that required close attention. We deeply investigated how such difficulties were overcome in the literature, revealing that crucial arguments for the correctness proof of the existing algorithms have been neglected. The systematic lack of rigorous arguments with respect to necessary conditions required for providing correctness proofs deeply affects the The work has been supported in part by the European project "Geospatial based Environment for Optimisation Systems Addressing Fire Emergencies" (GEO-SAFE), contract no. H2020-691161, and by the Italian National Group for Scientific Computation (GNCS-INdAM). Here we design a new deterministic distributed algorithm that fully characterizes APF showing its equivalence with the well-known Leader Election problem in the asynchronous model without chirality. Our approach is characterized by the use of logical predicates in order to formally describe our algorithm as well as its correctness. In addition to the relevance of our achievements, our techniques might help in revising previous results.

In this paper we present new results on the performance of the Minimum Spanning Tree heuristic for the Minimum Energy Broadcast Routing (MEBR) problem. We first prove that, for any number of dimensions d ≥ 2, the approximation ratio of the heuristic does not increase when the power attenuation coefficient α, that is the exponent to which the coverage distance must be raised to give the emission power, grows. Moreover, we show that, for any fixed instance, as a limit for α going to infinity, the ratio tends to the lower bound of Clementi et al. (Proceedings of the 18th annual symposium on theoretical aspects of computer science (STACS), pp. 121-131, 2001), Wan et al. (Wirel. Netw. 8(6):607-617, 2002) given by the d-dimensional kissing number, thus closing the existing gap between the upper and the lower bound. We then introduce a new analysis allowing to establish a 7.45-approximation ratio for Algorithmica (2007) 49: 318-336 319 the 2-dimensional case, thus significantly decreasing the previously known 12 upper bound (Wan et al. in Wirel. Netw. 8(6):607-617, 2002) (actually corrected to 12.15 in Klasing et al. (Proceedings of the 3rd IFIP-TC6 international networking conference, pp. 866-877, 2004)). Finally, we extend our analysis to any number of dimensions d ≥ 2 and any α ≥ d, obtaining a general approximation ratio of 3 d − 1, again independent of α. The improvements of the approximation ratios are specifically significant in comparison with the lower bounds given by the kissing numbers, as these grow at least exponentially with respect to d.

Distributed greedy coloring is an interesting and intuitive variation of the standard coloring problem. Given an order among the colors, a coloring is said to be greedy if there does not exist a vertex for which its associated color can be replaced by a color of lower position in the fixed order without violating the property that neighboring vertices must receive different colors. We consider the problems of Greedy Coloring and Largest First Coloring (a variant of greedy coloring with strengthened constraints) in the Linial model of distributed computation, providing lower and upper bounds and a comparison to the ( + 1)-Coloring and Maximal Independent Set problems, with being the maximum vertex degree in G.

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