Most Likely Voronoi Diagrams in Higher Dimensions

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 Omega(n^{2d}) where n is the number of points. This settles an open question raised in a recent (ISAAC 2014) paper of Suri and Verbeek, 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 ~O(n^{ceil{d/2}}) where the ~O(*) means that logarithmic factors are suppressed. In the worst case, this bound is tight up to polylog factors.

[1]  Mark de Berg,et al.  Separability of imprecise points , 2014, Comput. Geom..

[2]  Charu C. Aggarwal,et al.  Managing and Mining Uncertain Data , 2009, Advances in Database Systems.

[3]  Subhash Suri,et al.  Hyperplane Separability and Convexity of Probabilistic Point Sets , 2016, Symposium on Computational Geometry.

[4]  Subhash Suri,et al.  On the Most Likely Voronoi Diagram and Nearest Neighbor Searching , 2016, Int. J. Comput. Geom. Appl..

[5]  Jeff M. Phillips,et al.  Range counting coresets for uncertain data , 2013, SoCG '13.

[6]  Pankaj K. Agarwal,et al.  Convex Hulls Under Uncertainty , 2016, Algorithmica.

[7]  Haim Kaplan,et al.  The Overlay of Minimization Diagrams in a Randomized Incremental Construction , 2011, Discret. Comput. Geom..

[8]  Subhash Suri,et al.  Containment and Evasion in Stochastic Point Data , 2016, LATIN.

[9]  Maarten Löffler,et al.  Geometric Computations on Indecisive and Uncertain Points , 2012, ArXiv.

[10]  Sariel Har-Peled,et al.  On the Complexity of Randomly Weighted Voronoi Diagrams , 2014, SoCG.

[11]  Pankaj K. Agarwal,et al.  Range searching on uncertain data , 2012, TALG.

[12]  Timothy M. Chan,et al.  Stochastic minimum spanning trees in euclidean spaces , 2011, SoCG '11.

[13]  Philip S. Yu,et al.  A Survey of Uncertain Data Algorithms and Applications , 2009, IEEE Transactions on Knowledge and Data Engineering.

[14]  Timothy M. Chan,et al.  Closest pair and the post office problem for stochastic points , 2011, Comput. Geom..

[15]  Subhash Suri,et al.  Hyperplane separability and convexity of probabilistic point sets , 2017, J. Comput. Geom..

[16]  Yuan Li,et al.  On the arrangement of stochastic lines in ℤ2 , 2017, J. Discrete Algorithms.

[17]  Pankaj K. Agarwal,et al.  Nearest-neighbor searching under uncertainty , 2012, PODS.

[18]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[19]  Hsien-Kuei Hwang,et al.  Maxima in hypercubes , 2005, Random Struct. Algorithms.

[20]  Micha Sharir,et al.  Arrangements and Their Applications , 2000, Handbook of Computational Geometry.

[21]  Yuan Li,et al.  On the Separability of Stochastic Geometric Objects, with Applications , 2016, SoCG.