Properties and an approximation algorithm of round-tour Voronoi diagrams
Transactions on computational science IX
Properties and an approximation algorithm of round-tour Voronoi diagrams
Transactions on computational science IX
On multiplicatively weighted Voronoi diagrams for lines in the plane
Transactions on computational science XIII
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he triangle-perimeter 2-site distance function defines the ``distance'' ${\PP}(x,(p,q))$ from a point $x$ to two other points $p,q$ as the perimeter of the triangle whose vertices are $x,p,q$.Accordingly, given a set $S$ of $n$ points in the plane, the Voronoi diagram of $S$ with respect to $\PP$, denoted as $V_{\PP}(S)$, is the subdivision of the plane into regions, where the region of $p,q \in S$ is the locus of all points closer to $p,q$ (according to $\PP$) than to any other pair of sites in $S$.In this paper we prove a theorem about the perimeters of triangles, two of whose vertices are on a given circle.We use this theorem to show that the combinatorial complexity of $V_{\PP}(S)$ is $O(n^{2+\eps})$ (for any $\eps 0$).Consequently, we show that one can compute $V_{\PP}(S)$ in $O(n^{2+\eps})$ time and space.