Minimum-cost broadcast through varying-size neighborcast

Abstract

In traditional multihop network broadcast problems, in which a message beginning at one node is efficiently relayed to all others, cost models typically used involve a charge for each unicast or each broadcast. These settings lead to a minimum spanning tree (MST) problem or the Connected Dominating Set (CDS) problem, respectively. Neglected, however, is the study of intermediate models in which a node can choose to transmit to an arbitrary subset of its neighbors, at a cost based on the number of recipients (due e.g. to acknowledgements or repeat transmissions). We focus in this paper on a transmission cost model of the form 1 + A k b , where k is the number of recipients, b ≥ 0, and A ≥ 0, which subsumes MST, CDS, and other problems. We give a systematic analysis of this problem as parameterized by b (relative to A), including positive and negative results. In particular, we show the problem is approximable with a factor varying from 2 + 2H Δ down to 2 as b varies from 0 to 1 (via a modified CDS algorithm), and thence with a factor varying from 2 to 1 (i.e., optimal) as b varies from 1 to , and optimal thereafter (both via spanning tree). For arbitrary cost functions of the form 1 + Af(k), these algorithms provide a 2 + 2H Δ-approximation whenever f(k) is sublinear and a (1 + A)/A-approximation whenever f(k) is superlinear, respectively. We also show that the problem is optimally solvable for any b when the network is a clique or a tree. © 2012 Springer-Verlag.