Higher-dimensional Voronoi diagrams in linear expected time

ABSTRACT

This work is the first to validate theoretically the suspicions of many researchers — that the “average” Voronoi diagram is combinatorially quite simple and can be constructed quickly. Specifically, assuming that dimension d is fixed, and that n input points are chosen independently from the uniform distribution on the unit d-ball, it is proved that

• the expected number of simplices of the dual of the Voronoi diagram is Θ(n) (exact constants are derived for the high-order term), and

• a relatively simple algorithm exists for constructing the Voronoi diagram in Θ(n) time.

It is likely that the methods developed in the analysis will be applicable to other related quantities and other probability distributions.

References

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Index Terms

• Mathematics of computing

  • Discrete mathematics

    • Combinatorics

    • Graph theory

  • Mathematical analysis

    • Differential equations

      • Ordinary differential equations

    • Numerical analysis

      • Computations in finite fields

      • Interpolation