The Online Broadcast Range-Assignment Problem

Abstract

AbstractLet $$P=\{p_0,\ldots ,p_{n-1}\}$$ P = { p 0 , … , p n - 1 } be a set of points in $${\mathbb R}^d$$ R d , modeling devices in a wireless network. A range assignment assigns a range $$r(p_i)$$ r ( p i ) to each point $$p_i\in P$$ p i ∈ P , thus inducing a directed communication graph $$\mathcal {G}_r$$ G r in which there is a directed edge $$(p_i,p_j)$$ ( p i , p j ) iff $${{\,\textrm{dist}\,}}(p_i, p_j) \leqslant r(p_i)$$ dist ( p i , p j ) ⩽ r ( p i ) , where $${{\,\textrm{dist}\,}}(p_i,p_j)$$ dist ( p i , p j ) denotes the distance between $$p_i$$ p i and $$p_j$$ p j . The range-assignment problem is to assign the transmission ranges such that $$\mathcal {G}_r$$ G r has a certain desirable property, while minimizing the cost of the assignment; here the cost is given by $$\sum _{p_i\in P} r(p_i)^{\alpha }$$ ∑ p i ∈ P r ( p i ) α , for some constant $$\alpha >1$$ α > 1 called the distance-power gradient. We introduce the online version of the range-assignment problem, where the points $$p_j$$ 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 $$\mathcal {G}_r$$ G r has a broadcast tree rooted at the first point $$p_0$$ p 0 . Our results include the following. We prove that already in $${\mathbb R}^1$$ R 1 , a 1-competitive algorithm does not exist. In particular, for distance-power gradient $$\alpha =2$$ α = 2 any online algorithm has competitive ratio at least 1.57. For points in $${\mathbb R}^1$$ R 1 and $${\mathbb R}^2$$ R 2 , we analyze two natural strategies for updating the range assignment upon the arrival of a new point $$p_j$$ p j . The strategies do not change the assignment if $$p_j$$ 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 $${{\,\textrm{nn}\,}}(p_j)$$ nn ( p j ) , the nearest neighbor of $$p_j$$ p j , to $${{\,\textrm{dist}\,}}(p_j, {{\,\textrm{nn}\,}}(p_j))$$ dist ( p j , nn ( p j ) ) , and Cheapest Increase (ci) increases the range of the point $$p_i$$ p i for which the resulting cost increase to be able to reach the new point $$p_j$$ 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 $$\alpha $$ α . We also analyze the following variant of nn in $${\mathbb R}^2$$ R 2 for $$\alpha =2$$ α = 2 : 2-Nearest-Neighbor (2-nn) increases the range of $${{\,\textrm{nn}\,}}(p_j)$$ nn ( p j ) to $$2\cdot {{\,\textrm{dist}\,}}(p_j,{{\,\textrm{nn}\,}}(p_j))$$ 2 · dist ( p j , nn ( p j ) ) , We generalize the problem to points in arbitrary metric spaces, where we present an $$O(\log n)$$ O ( log n ) -competitive algorithm.