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arXiv · 1006.1921

Cloning Voronoi Diagrams via Retroactive Data Structures

Matthew T. Dickerson·David Eppstein·Michael T. Goodrich
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Abstract

We address the problem of replicating a Voronoi diagram $V(S)$ of a planar point set $S$ by making proximity queries, which are of three possible (in decreasing order of information content): 1. the exact location of the nearest site(s) in $S$; 2. the distance to and label(s) of the nearest site(s) in $S$; 3. a unique label for every nearest site in $S$. We provide algorithms showing how queries of Type 1 and Type 2 allow an exact cloning of $V(S)$ with $O(n)$ queries and $O(n \log^2 n)$ processing time. We also prove that queries of Type 3 can never exactly clone $V(S)$, but we show that with $O(n \log\frac{1}ε)$ queries we can construct an $ε$-approximate cloning of $V(S)$. In addition to showing the limits of nearest-neighbor database security, our methods also provide one of the first natural algorithmic applications of retroactive data structures.

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Matthew T. Dickerson, David Eppstein, Michael T. Goodrich. 2010-06-09. Cloning Voronoi Diagrams via Retroactive Data Structures. https://arxiv.org/abs/1006.1921

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