Greedy Drawings of Triangulations

Abstract

Greedy Routing is a class of routing algorithms in which the packets are forwarded in a manner that reduces the distance to the destination at every step. In an attempt to provide theoretical guarantees for a class of greedy routing algorithms, Papadimitriou and Ratajczak (Theor. Comput. Sci. 344(1):3–14, 2005) came up with the following conjecture: Any 3-connected planar graph can be drawn in the plane such that for every pair of vertices s and t a distance decreasing path can be found. A path s=v 1,v 2,…,v k<=t in a drawing is said to be distance decreasing if ‖v i−t‖<‖v i−1−t‖,2≤i≤k where ‖…‖ denotes the Euclidean distance. We settle this conjecture in the affirmative for the case of triangulations. A partitioning of the edges of a triangulation G into 3 trees, called the realizer of G, was first developed by Schnyder who also gave a drawing algorithm based on this. We generalize Schnyder’s algorithm to obtain a whole class of drawings of any given triangulation G. We show, using the Knaster–Kuratowski–Mazurkiewicz Theorem, that some drawing of G belonging to this class is greedy.