## A compact piecewise-linear Voronoi diagram for convex sites in the plane (1996)

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Venue: | Discrete Comput. Geom |

Citations: | 27 - 4 self |

### BibTeX

@INPROCEEDINGS{Mcallister96acompact,

author = {Michael Mcallister and David Kirkpatrick and Jack Snoeyink},

title = {A compact piecewise-linear Voronoi diagram for convex sites in the plane},

booktitle = {Discrete Comput. Geom},

year = {1996},

pages = {15--73}

}

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### Abstract

In the plane, the post-ofice problem, which asks for the closest site to a query site, and retraction motion planning, which asks for a one-dimensional retract of the free space of a robot, are both classt-cally solved by computing a Voronoi diagram. When the sites are k disjoint convex sets, we give a com-pact representation of the Voronoi diagram, using O(k) line segments, that is suficient for logarithmic time post-ofice location queries and motion plan-ning. If these sets are polygons with n total vertices, we compute this diagram optimally in O ( k log n) de-terministic time for the Euclidean metric and in O(k logn logm) deterministic time for the convex distance function defined by a convex m-gon. 1