Greedy drawings of triangulations

• Authors: Raghavan Dhandapani
• Venue: Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
• Year: 2008

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 [19] 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 = v1, v2,…,vk=t in a drawing is said to be distance decreasing if ||vi - t|| vi-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 Walter 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.