Faster approximation for Symmetric Min-Power Broadcast

Given a directed simple graph G = (V,E) and a cost function c : E → R+, the power of a vertex u in a directed spanning subgraph H is given by pH(u) = maxuv∈E(H) c(uv), and corresponds to the energy consumption required for the wireless node u to transmit to all nodes v with uv ∈ E(H). The power of H is given by p(H) = ∑ u∈V pH(u). Power Assignment seeks to minimize p(H) while H satisfies some connectivity constraint. In this paper, we assume E is bidirected (for every directed edge e ∈ E, the opposite edge exists and has the same cost), a “source” y ∈ V is also given as part of the input, and H is required to contain a directed path from y to every vertex of V . This is the NP-Hard Symmetric Min-Power Broadcast problem. In terms of approximation, it is known that one cannot obtain a ratio better than ln |V |, and at least five algorithms with approximation ratio O(ln |V |) have been published from 2002 to 2007. Here we take one of them, the 2(1+ln |V |)-approximation of Fredrick Mtenzi and Yingyu Wan, and improve its running time from O(|V ||E|) to O(|E| log |V |), by careful bookkeeping and by using a previously-known geometry-based data structure.