Approximating Multiplicatively Weighted Voronoi Diagrams: Efficient Construction with Linear Size

TL;DR

Novel approximation algorithms that output a cube-based subdivision such that the weighted distance of a point with respect to the associated site is at most $(1+\varepsilon)$ times the minimum weighted distance, for any fixed parameter $\varepsilon \in (0,1)$.

Abstract

Given a set of $n$ sites from $\mathbb{R}^d$, each having some positive weight factor, the Multiplicatively Weighted Voronoi Diagram is a subdivision of space that associates each cell to the site whose weighted Euclidean distance is minimal for all points in the cell. We give novel approximation algorithms that output a cube-based subdivision such that the weighted distance of a point with respect to the associated site is at most $(1+\varepsilon)$ times the minimum weighted distance, for any fixed parameter $\varepsilon \in (0,1)$. The diagram size is $O_d(n \log(1/\varepsilon)/\varepsilon^{d-1})$ and the construction time is within an $O_D(\log(n)/\varepsilon^{(d+5)/2})$-factor of the size bound. We also prove a matching lower bound for the size, showing that the proposed method is the first to achieve \emph{optimal size}, up to $Θ(1)^d$-factors. In particular, the obscure $\log(1/\varepsilon)$ factor is unavoidable. As a by-product, we obtain a factor $d^{O(d)}$ improvement in size for the unweighted case and $O(d \log(n) + d^2 \log(1/\varepsilon))$ point-location time in the subdivision, improving the known query bound by one $d$-factor. The key ingredients of our approximation algorithms are the study of convex regions that we call cores, an adaptive refinement algorithm to obtain optimal size, and a novel notion of \emph{bisector coresets}, which may be of independent interest. In particular, we show that coresets with $O_d(1/\varepsilon^{(d+3)/2})$ worst-case size can be computed in near-linear time.

Figures (5)

• Figure 1: The top shows an example of an exact MWVD of five sites ($\varepsilon_S=0$). The bottom shows an $\varepsilon_S$-AMWVD of the same instance obtained from cores with $\varepsilon_S=0.01$. Result squares of the proposed Adaptive Refinement algorithm (Section \ref{['sec:core_algorithm']}) for all four cores are shown as black overlay.
• Figure 2: Cases (HH), (LH), (HL), and (LL), for covering an absent ball $(i,j) \in B_i\setminus A_i$.
• Figure 4: Illustration of the proof of Lemma \ref{['lem:rot']}.
• Figure 5: Illustration of the proof of Lemma \ref{['lem:trans-heavy']}.
• Figure 6: The values $t^*_{ij}$ in $[a,b]$ are partitioned by intervals $I_0,\ldots,I_m$ of length $a \varepsilon_C/2$.

Theorems & Definitions (18)

• Lemma 0
• Definition 2: Distance Classes
• Lemma 3: Simple Bound
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma 7: Rotations at $s_i$
• Lemma 7: Translations
• Lemma 8: Constant per cone
• Theorem 9