On Succinct Convex Greedy Drawing of 3-Connected Plane Graphs

Abstract

Geometric routing by using virtual locations is an ele gant way for solving network routing problems. In its simplest form, greedy routing, a message is simply for warded to a neighbor that is closer to the destination. It has been an open conjecture whether every 3-connected plane graph has a greedy drawing in R2 (by Papadimitriou and Ratajczak [23]). Leighton and Moitra [20] recently settled this conjecture positively. One main drawback of this approach is that the coordinates of the virtual locations requires Ω(n log n) bits to repre sent (the same space usage as traditional routing table approaches). This makes greedy routing infeasible in applications. A similar result was obtained by Angelini et al. [2]. However, neither of the two papers give the time efficiency analysis of their algorithms. In addition, as pointed out in [16], the drawings in these two papers are not necessarily planar nor convex. In this paper, we show that the classical Schnyder drawing in R2 of plane triangulations is greedy with respect to a simple natural metric function H(u, v) over R2 that is equivalent to Euclidean metric DE(u, v) (in the sense that DE(u,v) < H(u, v) ≤ 2√2 DE(u, v).) The drawing is succinct, using two integer coordinates between 0 and 2n − 5. For 3-connected plane graphs, there is another conjecture by Papadimitriou and Ratajczak (as stated in [16]): Convex Greedy Embedding Conjecture: Every 3-connected planar graph has a convex greedy embedding in the Euclidean plane. In a recent paper [6], Cao et al. provided a plane graph G and showed that any convex greedy embedding of G in Euclidean plane must use Ω(n)-bit coordinates Thus, if we add the succinctness requirement, the Convex Greedy Embedding Conjecture is false In this paper, we show that the classical Schnyder drawing in R2 of 3-connected plane graphs is weakly greedy with respect to the same metric function H(*, *) The drawing is planar, convex, and succinct, using two integer coordinates between 0 and f (where f is the number of internal faces of G).