Dynamic updates of succinct triangulations

Luca Castelli Aleardi and Olivier Devillers and Gilles Schaeffer

Abstract:

In a recent article, we presented a succinct representation of triangulations that supports efficient navigation operations. Here this representation is improved to allow for efficient local updates of the triangulations.

Precisely, we show how a succinct representation of a triangulation with $m$ triangles can be maintained under vertex insertions in $O(1)$ amortized time and under vertex deletions/edge flips in $O(\lg^{2} m)$ amortized time.

Our structure achieves the information theory bound for the storage for the class of triangulations with a boundary, requiring asymptotically $2.17m+o(m)$ bits, and supports adjacency queries between triangles in $O(1)$ time (an extra amount of $O(g\lg m)$ bits are needed for representing triangulations of genus $g$ surfaces).