Radio networks are a long-studied model for distributed system of devices which communicate wirelessly. When these devices are mobile or have limited capabilities, the system is often best modeled by the ad-hoc variant, in which the devices do not know the structure of the network. A large body of work has been devoted to designing algorithms for the ad-hoc model, particularly for fundamental communications tasks such as broadcasting. Most of these algorithms, however, assume that devices have some network knowledge (usually bounds on the number of nodes in the network @math , and the diameter @math ), which may not always be realistic in systems with weak devices or gradual deployment. Very little is known about what can be done when this information is not available. This is the issue we address in this work, by presenting the first broadcasting algorithms for networks in which nodes have no prior knowledge whatsoever. We demonstrate that lack of parameter knowledge can be overcome at only a small increase in running time. Specifically, we show that in networks without collision detection, broadcast can be achieved in @math time, almost reaching the @math lower bound. We also give an algorithm for directed networks with collision detection, which requires only @math time.
Deterministic algorithms for broadcasting have also been studied; for undirected networks the fastest known algorithm is the @math -time algorithm of @cite_7 , while for directed networks it is the @math -time algorithm of @cite_3 .
@cite_3 @cite_7
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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 G=(V,E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive one-sided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of [SIAM J. Comput., 28 (1999), pp. 1167--1181], we describe an @math -time algorithm APASP2 for computing all distances in G with an additive one-sided error of at most 2. Algorithm APASP2 is simple, easy to implement, and faster than the fastest known matrix-multiplication algorithm. Furthermore, for every even k>2, we describe an @math -time algorithm APASPk for computing all distances in G with an additive one-sided error of at most k. We also give an @math -time algorithm @math for producing stretch 3 estimated distances in an unweighted and undirected graph on n vertices. No constant stretch factor was previously achieved in @math time. We say that a weighted graph F=(V,E') k-emulates an unweighted graph G=(V,E) if for every @math we have @math . We show that every unweighted graph on n vertices has a 2-emulator with @math edges and a 4-emulator with @math edges. These results are asymptotically tight. Finally, we show that any weighted undirected graph on n vertices has a 3-spanner with @math edges and that such a 3-spanner can be built in @math time. We also describe an @math -time algorithm for estimating all distances in a weighted undirected graph on n vertices with a stretch factor of at most 3. We present a method for solving the shortest transshipmen problem - also known as uncapacitated minimum cost flow - up to a multiplicative error of @math in undirected graphs with polynomially bounded integer edge weights using a tailored gradient descent algorithm. An important special case of the transshipment problem is the single-source shortest paths (SSSP) problem. Our gradient descent algorithm takes @math iterations, and in each iteration it needs to solve the transshipment problem up to a multiplicative error of @math , where @math is the number of nodes. In particular, this allows us to perform a single iteration by computing a solution on a sparse spanner of logarithmic stretch. As a consequence, we improve prior work by obtaining the following results: (1) Broadcast congest model: @math -approximate SSSP using @math rounds, where @math is the (hop) diameter of the network. (2) Broadcast congested clique model: @math -approximate transshipment and SSSP using @math rounds. (3) Multipass streaming model: @math -approximate transshipment and SSSP using @math space and @math passes. The previously fastest algorithms for these models leverage sparse hop sets. We bypass the hop set construction; computing a spanner is sufficient with our method. The above bounds assume non-negative integer edge weights that are polynomially bounded in @math ; for general non-negative weights, running times scale with the logarithm of the maximum ratio between non-zero weights. In case of asymmetric costs, running times scale with the maximum ratio between the costs of both directions over all edges. We consider distributed broadcasting in radio networks, modeled as undirected graphs, whose nodes have no information on the topology of the network, nor even on their immediate neighborhood. For randomized broadcasting, we give an algorithm working in expected time O(D log(n D) + log2 n) in n-node radio networks of diameter D, which is optimal, as it matches the lower bounds of [1] and Kushilevitz and Mansour [14]. Our algorithm improves the best previously known randomized broadcasting algorithm of Bar-Yehuda, Goldreich and Itai [3], running in expected time O(D log n + log2 n). For deterministic broadcasting, we show the lower bound Ω(n(log n) (log (n D)))) on broadcasting time in n-node radio networks of diameter D. This implies previously known lower bounds of Bar-Yehuda, Goldreich and Itai [3] and Bruschi and Del Pinto [5], and is sharper than any of them in many cases. We also give an algorithm working in time O(n log n), thus shrinking -- for the first time -- the gap between the upper and the lower bound on deterministic broadcasting time to a logarithmic factor.