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 $\log_2 (\frac{1}{A}+2)$, 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.

Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-09-2-0053. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation here on.