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Given a set of $n$ sites from $\mathbb{R}^d$, each having some positiveweight factor, the Multiplicative Weighted Voronoi Diagram is a subdivision ofspace that associates each cell to the site whose weighted Euclidean distanceis minimal for its points.  We give an approximation algorithm that outputs a subdivision such that theweighted 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 ${\cal O}(n\log(1/\varepsilon)/\varepsilon^{d-1})$ and the construction time is within afactor ${\cal O} (1/\varepsilon^{(d+1)d} +\log(n)/\varepsilon^{d+2} )$ of theoutput size. As a by-product, we obtain ${\cal O}(\log( n/\varepsilon))$point-location query time in the subdivision.  The key ingredients of the proposed method are the study of convex regionsthat we call cores, an adaptive refinement algorithm to obtain small outputsize, and a combination of Semi-Separated Pair Decompositions and conic spacepartitions to obtain efficient runtime.

Approximating Multiplicatively Weighted Voronoi Diagrams: Efficient Construction with Linear Size