# Most likely Voronoi diagrams in higher dimensions

## Abstract

The Most Likely Voronoi Diagram is a generalization of the well known Voronoi Diagrams to a stochastic setting, where a stochastic point is a point associated with a given probability of existence, and the cell for such a point is the set of points which would classify the given point as its most likely nearest neighbor. We investigate the complexity of this subdivision of space in d dimensions. We show that in the general case, the complexity of such a subdivision is Ω(n2d) where n is the number of points. This settles an open question raised in a recent (ISAAC 2014) paper of Suri and Verbeek [24], which first defined the Most Likely Voronoi Diagram. We also show that when the probabilities are assigned using a random permutation of a fixed set of values, in expectation the complexity is only Õ(n⌈d/2⌉) where the Õ(•) means that logarithmic factors are suppressed. In the worst case, this bound is tight up to polylog factors.

## Publication Title

Leibniz International Proceedings in Informatics, LIPIcs

## Recommended Citation

Kumar, N., Raichel, B., Suri, S., & Verbeek, K.
(2016). Most likely Voronoi diagrams in higher dimensions.* Leibniz International Proceedings in Informatics, LIPIcs**, 65*, 31.1-31.14.
https://doi.org/10.4230/LIPIcs.FSTTCS.2016.31