Succinct Greedy Drawings Do Not Always Exist ⋆ Patrizio Angelini, Giuseppe Di Battista, and Fabrizio Frati Dipartimento di Informatica e Automazione – Roma Tre University, Italy {angelini,gdb,frati}@dia.uniroma3.it Abstract. A greedy drawing is a graph drawing containing a distance-decreasing path for every pair of nodes. A path (v0,v1,...,vm) is distance-decreasing if d(vi ,vm) <d(vi-1,vm), for i =1,...,m. Greedy drawings easily support geographic greedy routing. Hence, a natural and practical problem is the one of constructing greedy drawings in the plane using few bits for representing vertex Cartesian coordinates and using the Euclidean distance as a metric. We show that there exist greedy-drawable graphs that do not admit any greedy drawing in which the Cartesian coordinates have less than a polynomial number of bits. 1 Introduction In geographic routing nodes forward packets based on their geographic locations. A very simple geographic routing protocol is greedy routing, in which each node knows its location, the location of its neighbors, and the location of the packet’s destination. Based on this information, a node forwards the packet to a neighbor that is closer than itself to the destination’s geographic location. Unfortunately, greedy routing has two weaknesses. First, GPS devices, typically used to determine coordinates, are expensive and increase the energy consumption of the nodes. Second, a bad interaction between the network topology and the node locations can lead to situations in which the communication fails because a void has been reached, i.e., a packet has reached a node whose neighbors are all farther from the destination than the node itself. A brilliant solution to the greedy routing weaknesses has been proposed by Rao et al., who in [13] proposed a protocol in which nodes are assigned virtual coordinates and the standard greedy routing algorithm is applied relying on such virtual locations rather than on the geographic coordinates. Clearly, virtual coordinates need not to reflect the nodes actual positions and, hence, they can be suitably chosen to guarantee that the greedy routing algorithm succeeds in delivering packets. After the publication of [13], intense research efforts have been devoted to deter- mine: (i) Which network topologies admit a virtual coordinates assignment such that greedy routing is guaranteed to work. (ii) Which distance metrics, which systems of coordinates, and how many dimensions are suitable for virtual coordinates. (iii) How many bits are needed to represent the vertex coordinates. From a graph-theoretical point of view, Problem (i) can be stated as follows: Which are the graphs that admit a greedy drawing, i.e., a drawing such that, for every two nodes ⋆ This work is partially supported by the Italian Ministry of Research, Grant number RBIP06BZW8, FIRB project “Advanced tracking system in intermodal freight transportation”. D. Eppstein and E.R. Gansner (Eds.): GD 2009, LNCS 5849, pp. 171–182, 2010. c  Springer-Verlag Berlin Heidelberg 2010