Motivated by topology control in ad hoc wireless networks, Power Assignment is a family of problems, each defined by a certain connectivity constraint (such as strong connectivity). The input consists of a directed complete weighted digraph G = ( V, c ) (that is, c : V × V → 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 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 the given connectivity constraint. Min-Power Bounded-Hops Broadcast is a power assignment problem which has as additional input a positive integer d and a r ∈ V. The output H must be a r-rooted outgoing arborescence of depth at most d. We give an ( O ( log n ), O ( log n ) ) bicriteria approximation algorithm for Min-Power Bounded-Hops Broadcast: that is, our output has depth at most O ( d log n ) and power at most O ( log n ) times the optimum solution. For the Euclidean case, when c ( u, v ) = c ( v, u ) = ∥ u, v ∥κ (here ∥ u, v ∥ is the Euclidean distance and κ is a constant between 2 and 5), the output of our algorithm can be modified to give a O ( ( log n )κ ) approximation ratio. Previous results for Min-Power Bounded-Hops Broadcast are only exact algorithms based on dynamic programming for the case when the nodes lie on the line and c ( u, v ) = c ( v, u ) = ∥ u, v ∥κ. Our bicriteria results extend to Min-Power Bounded-Hops Strong Connectivity, where H must have a path of at most d edges in between any two nodes. Previous work for Min-Power Bounded-Hops Strong Connectivity consists only of constant or better approximation for special cases of the Euclidean case. © 2005 Elsevier B.V. All rights reserved.
Calinescu, G., Kapoor, S., & Sarwat, M. (2006). Bounded-hops power assignment in ad hoc wireless networks. Discrete Applied Mathematics, 154(9), 1358–1371. https://doi.org/10.1016/j.dam.2005.05.034