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Results on Clustering of Imprecise Points and Higher Order Voronoi Tessellations

Saghafian, Morteza | 2024

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 56827 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Haji Mirsadeghi, Mir Omid; Abam, Mohammad Ali
  7. Abstract:
  8. We study the problem of preclustering a set $B$ of imprecise points in~$\Reals^d$: we wish to cluster the regions specifying the potential locations of the points such that, no matter where the points are located within their regions, the resulting clustering approximates the optimal clustering for those locations. We consider $k$-center, $k$-median, and $k$-means clustering, and obtain the various results. Then we study the higher order Voronoi Tessellations as an important concept in clustering area. Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in $\Reals^2$ to $\Reals^3$, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of $n \geq 5$ points in $\Reals^3$, the number of regions in the order-$k$ Voronoi tessellation is $N_{k−1} − \binom{k}{2}n + n$, for $1 \leq k \leq n − 1$, in which $N_{k−1}$ is the sum of Euler characteristics of these function’s first $k − 1$ sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-$k$ Voronoi tessellation. Finally, trying to find an efficient algorithm to compute the order-$k$ Voronoi Tessellation for a set of input points, we present some interesting partial results regarding weighted Voronoi Tessellations and the overlay of them.
  9. Keywords:
  10. Computational Geometry ; Clustering ; Voronoi Tessellation ; Approximate Algorithm ; Imprecise Points Clustering

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