Expected time analysis for Delaunay point location

Authors:
Luc Devroye;Christophe Lemaire;Jean-Michel Moreau
Affiliations:
School of Computer Science, McGill University, 3480 University Street, Montreal, Canada H3A 2K6;School of Computer Science, McGill University, 3480 University Street, Montreal, Canada H3A 2K6;School of Computer Science, McGill University, 3480 University Street, Montreal, Canada H3A 2K6
Venue:
Computational Geometry: Theory and Applications
Year:
2004

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Abstract

We consider point location in Delaunay triangulations with the aid of simple data structures. In particular, we analyze methods in which a simple data structure is used to first locate a point close to the query point. For points uniformly distributed on the unit square, we show that the expected point location complexities are Θ(√n) for the Green-Sibson rectilinear search, Θ(n1/3) for Jump and Walk, Θ(n1/4) for BinSearch and Walk (which uses a 1-dimensional search tree), Θ(n0.056...) for search based on a random 2-d tree, and Θ(log n) for search aided by a 2-d median tree.