A linear-time algorithm for broadcast domination in a tree

The broadcast domination problem is a variant of the classical minimum dominating set problem in which a transmitter of power p at vertex v is capable of dominating (broadcasting to) all vertices within distance p from v. Our goal is to assign a broadcast power f(v) to every vertex v in a graph such that ∑ v∈V f(v) is minimized, and such that every vertex u with f(u) = 0 is within distance f(v) of some vertex v with f(v) > 0. The problem is solvable in polynomial time on a general graph [15], and Blair et al. [6] gave an O(n) algorithm for trees. In this paper, we provide an O(n) algorithm for trees. Our algorithm is notable due to the fact that it makes decisions for each vertex v based on “non-local” information from vertices far away from v, whereas almost all other linear-time algorithms for trees only make use of local information.