The Online Broadcast Range-Assignment Problem

Abstract

Let P={p_0,ldots ,p_{n-1}} be a set of points in {mathbb R}^d, modeling devices in a wireless network. A range assignment assigns a range r(p_i) to each point p_iin P, thus inducing a directed communication graph mathcal {G}_r in which there is a directed edge (p_i,p_j) iff {{,textrm{dist},}}(p_i, p_j) leqslant r(p_i), where {{,textrm{dist},}}(p_i,p_j) denotes the distance between p_i and p_j. The range-assignment problem is to assign the transmission ranges such that mathcal {G}_r has a certain desirable property, while minimizing the cost of the assignment; here the cost is given by sum _{p_iin P} r(p_i)^{alpha }, for some constant alpha >1 called the distance-power gradient. We introduce the online version of the range-assignment problem, where the points p_j 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 mathcal {G}_r has a broadcast tree rooted at the first point p_0. Our results include the following.We prove that already in {mathbb R}^1, a 1-competitive algorithm does not exist. In particular, for distance-power gradient alpha =2 any online algorithm has competitive ratio at least 1.57.For points in {mathbb R}^1 and {mathbb R}^2, we analyze two natural strategies for updating the range assignment upon the arrival of a new point p_j. The strategies do not change the assignment if p_j 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 {{,textrm{nn},}}(p_j), the nearest neighbor of p_j, to {{,textrm{dist},}}(p_j, {{,textrm{nn},}}(p_j)), and Cheapest Increase (ci) increases the range of the point p_i for which the resulting cost increase to be able to reach the new point p_j is minimal. We give lower and upper bounds on the competitive ratio of these strategies as a function of the distance-power gradient alpha . We also analyze the following variant of nn in {mathbb R}^2 for alpha =2: 2-Nearest-Neighbor (2-nn) increases the range of {{,textrm{nn},}}(p_j) to 2cdot {{,textrm{dist},}}(p_j,{{,textrm{nn},}}(p_j)),We generalize the problem to points in arbitrary metric spaces, where we present an O(log n)-competitive algorithm.