Self-improving algorithms for delaunay triangulations

Authors:
Kenneth L. Clarkson;C. Seshadhri
Affiliations:
IBM Almaden Research Center, San Jose, CA, USA;Princeton University, Princeton, NJ, USA
Venue:
Proceedings of the twenty-fourth annual symposium on Computational geometry
Year:
2008

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Abstract

We study the problem of two-dimensional Delaunay triangulation in the self-improving algorithms model [1]. We assume that the n points of the input each come from an independent, unknown, and arbitrary distribution. The first phase of our algorithm builds data structures that store relevant information about the input distribution. The second phase uses these data structures to efficiently compute the Delaunay triangulation of the input. The running time of our algorithm matches the information-theoretic lower bound for the given input distribution, implying that if the input distribution has low entropy, then our algorithm beats the standard Ω(n log n) bound for computing Delaunay triangulations. Our algorithm and analysis use a variety of techniques: ε-nets for disks, entropy-optimal point-location data structures, linear-time splitting of Delaunay triangulations, and information-theoretic arguments.