Linear-time reconstruction of delaunay triangulations with applications

Abstract

Many of the computational geometers’ favorite data structures are planar graphs, canonically determined by a set of geometric data, that take Θ(n log n) time to compute. Examples include 2-d Delaunay triangulation, trapezoidations of segments, and constrained Voronoi diagrams, and 3-d convex hulls. Given such a structure, one can determine a permutation of the data in O(n) time such that the data structure can be reconstructed from the permuted data in O(n) time by a simple incremental algorithm. As a consequence, one can permute a data file to "hide" a geometric structure, such as a tertian model based on the Delaunay triangulation of a set of sampled points, without disrupting other applications. One can even include "importance" in the ordering so the incremental reconstruction produces approximate terrain models as the data is read or received. For the Delaunay triangulation~ we can also handle input in degenerate position, even though the data structures may no longer be canonically defined.