The Online Broadcast Range-Assignment Problem
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Let P={p_0,…,p_n-1} be a set of points in ℝ^d, modeling devices in a wireless network. A range assignment assigns a range r(p_i) to each point p_i∈ P, thus inducing a directed communication graph G_r in which there is a directed edge (p_i,p_j) iff dist(p_i, p_j) ≤ r(p_i), where 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 G_r has a certain desirable property, while minimizing the cost of the assignment; here the cost is given by ∑_p_i∈ P r(p_i)^α, for some constant α>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 G_r has a broadcast tree rooted at the first point p_0. Our results include the following. - For d=1, a 1-competitive algorithm does not exist. In particular, for α=2 any online algorithm has competitive ratio at least 1.57. - For d=1 and d=2, we analyze two natural strategies: Upon the arrival of a new point p_j, Nearest-Neighbor increases the range of the nearest point to cover p_j and Cheapest Increase increases the range of the point for which the resulting cost increase to be able to reach p_j is minimal. - We generalize the problem to arbitrary metric spaces, where we present an O(log n)-competitive algorithm.
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