We give a unified view to greedy geometric routing algorithms in ad hoc networks. For this, we first present a general form of greedy routing algorithm using a class of objective functions which are invariant under congruent transformations of a point set. We show that several known greedy routing algorithms such as Greedy Routing, Compass Routing, and Midpoint Routing can be regarded as special cases of the generalized greedy routing algorithm. In addition, inspired by the unified view of greedy routing, we propose three new greedy routing algorithms. We then derive a sufficient condition for our generalized greedy routing algorithm to guarantee packet delivery on every Delaunay graph. This condition makes it easier to check whether a given routing algorithm guarantees packet delivery, and it is closed under convex linear combination of objective functions. It is shown that Greedy Routing, Midpoint Routing, and the three new greedy routing algorithms proposed in this paper satisfy the sufficient condition, i.e., they guarantee packet delivery on Delaunay graphs. We also discuss merits and demerits of these methods.

This paper presents a new proof to a polynomial-time algorithm for determining whether a given embedded graph is a Delaunay graph, i. e. , whether it is topologically equivalent to a Delaunay triangulation. The problem of recognizing the Delaunay graph had long been open. Recently Hodgson et al. gave a combinatorial characterization of the Delaunay graph, and thus constructed the polynomial-time algorithm for recognizing the Delaunay graphs. Their proof is based on sophisticated discussions on hyperbolic geometry. On the other hand, this paper gives another and simpler proof based on primitive arguments on Euclidean geometry. Moreover, the algorithm is applied to study the distribution of non-Delaunay graphs.