2012 | OriginalPaper | Buchkapitel
Succinct Strictly Convex Greedy Drawing of 3-Connected Plane Graphs
verfasst von : Jiun-Jie Wang, Xin He
Erschienen in: Frontiers in Algorithmics and Algorithmic Aspects in Information and Management
Verlag: Springer Berlin Heidelberg
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Geometric routing by using virtual locations is an elegant way for solving network routing problems. Greedy routing, where a message is simply forwarded to a neighbor that is closer to the destination, is a simple form of geometric routing. Papadimitriou and Ratajczak conjectured that every 3-connected plane graph has a greedy drawing in the
$\mathcal R^2$
plane [10]. Leighton and Moitra settled this conjecture positively in [9]. However, their drawings have two major drawbacks: (1) their drawings are not necessarily planar; and (2) Ω(
n
log
n
) bits are needed to represent the coordinates of their drawings, which is too large for routing algorithms for wireless networks. Recently, He and Zhang [8] showed that every triangulated plane graph has a succinct (using
O
(log
n
) bit coordinates) greedy drawing in
$\mathcal R^2$
plane with respect to a metric function derived from Schnyder realizer. However, their method fails for 3-connected plane graphs. In this paper, we show that every 3-connected plane graph has drawing in the
$\mathcal R^2$
plane, that is succinct, planar, strictly convex, and is greedy with respect to a metric function based on parameters derived from Schnyder wood.