The online broadcast range-assignment problem

Let P = {p0, . . ., pn−1} be a set of points in R^d, modeling devices in a wireless network. A range assignment assigns a range r(pi) to each point pi ∈ P, thus inducing a directed communication graph Gr in which there is a directed edge (pi, pj) iff dist(pi, pj) 6 r(pi), where dist(pi, pj) denotes the distance between pi and pj. The range-assignment problem is to assign the transmission ranges such that Gr has a certain desirable property, while minimizing the cost of the assignment; here the cost is given by ^P_pi∈_P r(pi)^α, for some constant α > 1 called the distance-power gradient. We introduce the online version of the range-assignment problem, where the points pj arrive one by one, and the range assignment has to be updated at each arrival. Following the standard in online algorithms, resources given out cannot be taken away – in our case this means that the transmission ranges will never decrease. The property we want to maintain is that Gr has a broadcast tree rooted at the first point p0. Our results include the following. We prove that already in R¹, a 1-competitive algorithm does not exist. In particular, for distance-power gradient α = 2 any online algorithm has competitive ratio at least 1.57. For points in R¹ and R², we analyze two natural strategies for updating the range assignment upon the arrival of a new point pj. The strategies do not change the assignment if pj is already within range of an existing point, otherwise they increase the range of a single point, as follows: Nearest-Neighbor (nn) increases the range of nn(pj), the nearest neighbor of pj, to dist(pj, nn(pj)), and Cheapest Increase (ci) increases the range of the point pi for which the resulting cost increase to be able to reach the new point pj is minimal. We give lower and upper bounds on the competitive ratio of these strategies as a function of the distance-power gradient α. We also analyze the following variant of nn in R² for α = 2: 2-Nearest-Neighbor (2-nn) increases the range of nn(pj) to 2 · dist(pj, nn(pj)), We generalize the problem to points in arbitrary metric spaces, where we present an O(log n)competitive algorithm.