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arXiv · 0904.2776

Schnyder woods for higher genus triangulated surfaces, with applications to encoding

Luca Castelli Aleardi·Eric Fusy·Thomas Lewiner
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Abstract

Schnyder woods are a well-known combinatorial structure for plane triangulations, which yields a decomposition into 3 spanning trees. We extend here definitions and algorithms for Schnyder woods to closed orientable surfaces of arbitrary genus. In particular, we describe a method to traverse a triangulation of genus $g$ and compute a so-called $g$-Schnyder wood on the way. As an application, we give a procedure to encode a triangulation of genus $g$ and $n$ vertices in $4n+O(g \log(n))$ bits. This matches the worst-case encoding rate of Edgebreaker in positive genus. All the algorithms presented here have execution time $O((n+g)g)$, hence are linear when the genus is fixed.

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Luca Castelli Aleardi, Eric Fusy, Thomas Lewiner. 2009-04-17. Schnyder woods for higher genus triangulated surfaces, with applications to encoding. https://arxiv.org/abs/0904.2776

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