Labeling Schemes for Deterministic Radio Multi-Broadcast

Abstract

We consider the multi-broadcast problem in arbitrary connected radio networks consisting of n nodes. There are k designated source nodes for some fixed k ∈ {1, . . . , n}, and each source node has a distinct piece of information that it wants to share with all nodes in the network. When k = 1, this is known as the broadcasting problem, and when k = n, this is known as the gossiping problem. We consider the feasibility of solving multi-broadcast deterministically in radio networks. It is known that multi-broadcast is solvable when the nodes have distinct identifiers (e.g., using round-robin), and, it has been shown by Ellen, Gorain, Miller, and Pelc (2019) that the broadcasting problem is solvable if the nodes have been carefully assigned 2bit labels rather than distinct identifiers. We set out to determine the shortest possible labels so that multi-broadcast can be solved deterministically in the labeled radio network by some universal deterministic distributed algorithm.First, we show that every radio network G with maximum degree ∆ can be labeled using O(min{log k, log ∆})-bit labels in such a way that multi-broadcast with k sources can be accomplished. This bound is tight, in the sense that there are networks such that any labeling scheme sufficient for multi-broadcast with k sources requires Ω(min{log k, log ∆})-bit labels.However, the above result is somewhat unsatisfactory: the bound is tight for certain network topologies (e.g., complete graphs), but there are networks where significantly shorter labels are sufficient. For example, we show how to construct a tree with maximum degree Θ( √ n) in which gossiping (i.e., multi-broadcast with n sources) can actually be solved after labeling the nodes with O(1)-bit labels. So, we set out to find a labeling scheme for multi-broadcast that uses the optimal number of distinct labels in every network. For all trees, we provide a labeling scheme and accompanying algorithm that will solve gossiping, and, we prove an impossibility result that demonstrates that our labeling scheme is optimal for gossiping in each tree. In particular, we prove that Θ(log D(G))-bit labels are necessary and sufficient in every tree G, where D(G) denotes the distinguishing number of G. This result also applies more generally to multi-broadcast in trees with k ∈ {2, . . . , n} sources in the case where the k sources are not known when the labeling scheme is applied.