An Analysis of Planarity in Face-Routing

Abstract

In this report we investigate the limits of routing according to left- or right-hand rule (LHR). Using LHR, a node upon receipt of a message will forward to the neighbour that sits next in counter-clockwise order in the network graph. When used to recover from greedy routing failures, LHR guarantees success if implemented over planar graphs. This is often referred to as face or geographic routing. In the current body of knowledge it is known that if planarity is violated then LHR is guaranteed only to eventually return to the point of origin. Our work seeks to understand why a non-planar environment stops LHR from making delivery guarantees. Our investigation begins with an analysis to enumerate all node con gurations that cause intersections. A trace over each con guration reveals that LHR is able to recover from all but a single case, the `umbrella' con guration so named for its appearance. We use this information to propose the Prohibitive Link Detection Protocol (PDLP) that can guarantee delivery over non-planar graphs using standard face-routing techniques. As the name implies, the protocol detects and circumvents the `bad' links that hamper LHR. The goal of this work is to maintain routing guarantees while disturbing the network graph as little as possible. In doing so, a new starting point emerges from which to build rich distributed protocols in the spirit of protocols such as CLDP and GDSTR.