A localized link removal and addition based planarization algorithm

Abstract

In wireless networks, the planar embedding of the communication network is often used in applications such as topology control and routing. Typically, wireless networks are nonplanar and hence algorithms which use local information are often applied to create planar graphs. These algorithms provably yield a connected planar graph as long as the network obeys certain assumptions like Unit Disk Graph (UDG). In this paper we propose a new explicit localized graph planarization algorithm to planarize graphs that are more general than UDGs, i.e. graphs satisfying redundancy property and a new property introduced by us called coexistence property. This algorithm detects intersections locally and planarizes networks by removing them. A link addition phase ensures that the network remains connected after the planarization algorithm. Theoretical analysis shows that our algorithm provably creates planar graphs without disconnecting the network graph if it has redundancy and coexistence properties. We also show these planar graphs are weak c-spanners at least in UDG modeled networks. Empirical analysis shows that they are as good as the best state of the art localized planarization algorithm with respect to the spanning ratio. © 2012 Springer-Verlag.