The Online Broadcast Range-Assignment Problem

TL;DR

This paper studies an online wireless network range assignment problem, providing bounds and algorithms for maintaining broadcast connectivity with minimal power cost as devices arrive.

Contribution

It introduces the first online algorithms for the range-assignment problem, analyzing their competitiveness and extending to arbitrary metric spaces.

Findings

No 1-competitive algorithm exists for d=1.

For d=1 and d=2, two natural strategies are analyzed.

An O(log n)-competitive algorithm is presented for general metric spaces.

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 $p_i\in P$, thus inducing a directed communication graph $G_r$ in which there is a directed edge $(p_i,p_j)$ iff $\textrm{dist}(p_i, p_j) \leq r(p_i)$, 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 $G_r$ has a certain desirable property, while minimizing the cost of the assignment; here the cost is given by $\sum_{p_i\in P} r(p_i)^{\alpha}$, 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 $G_r$ 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 $\alpha=2$ 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.