Efficient Graph Planarization in Sensor Networks and Local Routing Algorithm

Abstract

Abstract-In this paper, we propose an efficient planarization algorithm and a routing algorithm dedicated to Unit Disk Graphs whose nodes are localized using the Virtual Raw Anchor Coordinate system (VRAC). Our first algorithm computes a planar 2-spanner under light constraints on the edge lengths and induces a total exchange of at most 6n node identifiers. Its total computational complexity is O(n∆), with ∆ the maximum degree of the communication graph. The second algorithm that we present is a simple and efficient algorithm to route messages in this planar graph that requires routing tables with only three entries. We support these theoretical results by simulations showing the robustness of our algorithms when the coordinates are inaccurate.

Combinatorial Approach for Geographic Routing with Delivery Guarantees

“…We proved in (Huc et al, 2012) that under some conditions on the length of the edges the resulting graph G is planar and we discuss the stretch factor compared to the original graph G. Our aim in the following is to present a geographic routing algorithm on top of this planar graph that guarantee the delivery of the data. In the paper (Huc et al, 2012) we investigate some geometrical properties that imply that the graph G = ( V , E) is planar.…”

“…In the paper (Huc et al, 2012) we investigate some geometrical properties that imply that the graph G = ( V , E) is planar. Actually, the aim of these geometrical properties is to ensure that the following property is satisfied.…”

“…In our setting, the nodes are localized in the VRAC coordinate system and this associates to each node u the coordinates Figure 1. Moreover, we assume that all the nodes are inside the triangular region, A 1 A 2 A 3 , with suits the anchors A 1 , A 2 , A 3 , see (Huc et al, 2012) for the explanation of why this is necessary and the description of possible further works to remove this assumption. We can define three order relations on the set of nodes V .…”

“…In (Huc et al, 2012) we call the edges (u, v) with v in s u 1 , s u 3 , or s u 5 outgoing edges. With this terminology, a node has at most three outgoing edges and possibly many ingoing edges (and edge (u, v) such that v ∈ s u 2 , s u 4 or v ∈ s u 6 ).…”

“…The local planarization of the communication graph in the VRAC coordinate system is presented in (Huc et al, 2012) where we adapt the planarity criterion introduced in (Schnyder, 1989) to Unit Disk Graph (UDG), i.e. the nodes are positioned in a region of the plane and two nodes are connected if and only if the distance among them is less that a constant r called the range of communications.…”