# Voronoi diagrams in the moscow metric

## Abstract

Most of the streets of Moscow are either radii emanating from the Kremlin, or pieces of circles around it. We show that Voronoi diagrams for *n* points based on this metric can be computed in optimal *O*(*n* log *n*) time and linear space. To this end, we prove a general theorem stating that bisectors of suitably separated point sets do not contain loops if, beside other properties, there are no holes in the circles of the underlying metric. Then the Voronoi diagrams can be computed within *O*(*n* log *n*) steps, using a divide-and-conquer algorithm. Our theorem not only applies to the Moscow metric but to a large class of metrics including the symmetric convex distance functions and all composite metrics obtained by assigning the *L*_{1} or the *L*_{2} metric to the regions of a planar map.

## Keywords

Bisector computational geometry convex distance function metric norm robotics shortest path Voronoi diagram## Preview

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