Approximation algorithms for the mobile piercing set problem with applications to clustering in ad-hoc networks

Abstract

The main contributions of this paper are two-fold. First, we present a simple, general framework for obtaining efficient constant-factor approximation algorithms for the mobile piercing set (MPS) problem on unit-disks for standard metrics in fixed dimension vector spaces. More specifically, we provide low constant approximations for L1- and L∞-norms on a d-dimensional space, for any fixed d > 0, and for the L2-norm on 2- and 3-dimensional spaces. Our framework provides a family of fully-distributed and decentralized algorithms, which adapts (asymptotically) optimally to the mobility of disks, at the expense of a low degradation on the best known approximation factors of the respective centralized algorithms: Our algorithms take O(1) time to update the piercing set maintained, per movement of a disk. We also present a family of fully-distributed algorithms for the MPS problem which either match or improve the best known approximation bounds of centralized algorithms for the respective norms and dimensions.Second, we show how the proposed algorithms can be directly applied to provide theoretical performance analyses for two popular 1-hop clustering algorithms in ad-hoc networks: the lowest-id algorithm and the Least Cluster Change (LCC) algorithm. More specifically, we formally prove that the LCC algorithm adapts in constant time to the mobility of the network nodes, and minimizes (up to low constant factors) the number of 1-hop clusters maintained; we propose an alternative algorithm to the lowest-id algorithm which achieves a better approximation factor without increasing the cost of adapting to changes in the network topology. While there is a vast literature on simulation results for the LCC and the lowest-id algorithms, these had not been formally analysed prior to this work. We also present an O(log n)-approximation algorithm for the mobile piercing set problem for nonuniform disks (i.e., disks that may have different radii), with constant update time.