Cloning Voronoi Diagrams via Retroactive Data Structures

TL;DR

This paper explores how to replicate Voronoi diagrams using proximity queries, demonstrating exact cloning with certain query types and approximate cloning with others, while also applying retroactive data structures.

Contribution

It introduces algorithms for exact and approximate cloning of Voronoi diagrams using different query types and applies retroactive data structures to geometric problems.

Findings

Exact cloning with Type 1 and 2 queries in O(n) queries and O(n log^2 n) time

Type 3 queries cannot exactly clone V(S)

Approximate cloning achievable with O(n log(1/ε)) queries

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 $O(n \log\frac{1}{\epsilon})$ queries we can construct an $\epsilon$-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.