Minimum-power multicast routing in static ad hoc wireless networks

Wieselthier et al. (2000) proposed three greedy heuristics for Min-Power Asymmetric Broadcast Routing: SPT (shortestpath tree), MST (minimum spanning tree), and BIP (broadcasting incremental power). Wan et al. (2001) proved that SPT has an approximation ratio of at least (n/2) where n is the total number of nodes, and both MST and BIP have constant approximation ratios. Based on the approach of pruning, Wieselthier et al. also proposed three greedy heuristics for MinPower Asymmetric Multicast Routing: P-SPT (pruned shortest-path tree), P-MST (pruned minimum spanning tree), and PBIP (pruned broadcasting incremental power). In this paper, we first prove that the approximation ratios of these three heuristics are at least (n-1/2),n-1, and n-2-o(1), respectively. We then present constant-approxiation algorithms for MinPower Asymmetric Multicast Routing. We show that any (rho)-approximation Steiner tree algorithm gives rise to a c(rho)approximation heuristic for Min-Power Asymmetric Multicast Routing, where c is a constant between 6 and 12. In particular, the Takahashi-Matsuyama Steiner tree heuristic leads to a heuristic called SPF (shortest-path first), which has an approximation ratio of at most 2c. We also present another heuristic, called MIPF (minimum incremental path first), for MinPower Asymmetric Multicast Routing and show that its approximation ratio is between (13/3) and 2c. Both SPF and MIPF can be regarded as an adaptation of MST and BIP, respectively, in a different manner than pruning. Finally, we prove that any (rho)-approximation Steiner tree algorithm also gives rise to a 2(rho)-approximation algorithm for Min-Power Symmetric Multicast Routing.