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

TL;DR

This paper generalizes Schnyder woods from planar to higher genus surfaces, providing algorithms for their computation and an efficient encoding scheme for triangulations of arbitrary genus.

Contribution

It introduces a method to extend Schnyder woods to higher genus surfaces and develops an encoding procedure matching known bounds.

Findings

Encoding triangulations of genus g in 4n + O(g log n) bits.

Algorithms run in O((n+g)g) time, linear for fixed genus.

Provides a traversal method to compute g-Schnyder woods.

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.