Approximation algorithms for the mobile piercing set problem with applications to clustering,

Abstract. The main contributions of this paper are two-fold. First, we present a simple, general framework for obtaining efficient constantfactor 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 L 1 and L ∞ norms on a d-dimensional space, for any fixed d > 0, and for the L 2 norm on two-and three-dimensional spaces. Our framework provides a family of fully-distributed and decentralized algorithms, which adapt (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 space 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. While there is a vast literature on simulation results for the LCC and the lowest-id algorithms, these had not been formally analyzed 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

Approximation Algorithms for the Mobile Piercing Set Problem with Applications to Clustering in Ad-hoc Networks

The main contributions of this paper are two-fold. First, we present a simple, general framework for obtaining ecient constant-factor approximation algorithms for the mobile piercing set (MPS) problem on unit-disks for standard metrics in xed dimension vector spaces. More speci cally, we provide low constant approximations for L 1 and L1 norms on a d-dimensional space, for any xed d > 0, and for the L 2 norm on two- and three-dimensional spaces. Our framework provides a family of fullydistributed 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 space dimensions

Clustering and Hybrid Routing in Mobile Ad Hoc Networks

This dissertation focuses on clustering and hybrid routing in Mobile Ad Hoc Networks (MANET). Specifically, we study two different network-layer virtual infrastructures proposed for MANET: the explicit cluster infrastructure and the implicit zone infrastructure. In the first part of the dissertation, we propose a novel clustering scheme based on a number of properties of diameter-2 graphs to provide a general-purpose virtual infrastructure for MANET. Compared to virtual infrastructures with central nodes, our virtual infrastructure is more symmetric and stable, but still light-weight. In our clustering scheme, cluster initialization naturally blends into cluster maintenance, showing the unity between these two operations. We call our algorithm tree-based since cluster merge and split operations are performed based on a spanning tree maintained at some specific nodes. Extensive simulation results have shown the effectiveness of our clustering scheme when compared to other schemes proposed in the literature. In the second part of the dissertation, we propose TZRP (Two-Zone Routing Protocol) as a hybrid routing framework that can balance the tradeoffs between pure proactive, fuzzy proactive, and reactive routing approaches more effectively in a wide range of network conditions. In TZRP, each node maintains two zones: a Crisp Zone for proactive routing and efficient bordercasting, and a Fuzzy Zone for heuristic routing using imprecise locality information. The perimeter of the Crisp Zone is the boundary between pure proactive routing and fuzzy proactive routing, and the perimeter of the Fuzzy Zone is the boundary between proactive routing and reactive routing. By adjusting the sizes of these two zones, a reduced total routing control overhead can be achieved