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We address the problem of replicating a Voronoi diagram $V(S)$ of a planarpoint set $S$ by making proximity queries, which are of three possible (indecreasing order of information content): 1. the exact location of the nearestsite(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 showinghow 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 Type3 can never exactly clone $V(S)$, but we show that with $O(n\log\frac{1}{\epsilon})$ queries we can construct an $\epsilon$-approximatecloning of $V(S)$. In addition to showing the limits of nearest-neighbordatabase security, our methods also provide one of the first naturalalgorithmic applications of retroactive data structures.

Cloning Voronoi Diagrams via Retroactive Data Structures