Dilation, smoothed distance, and minimization diagrams of convex functions

Clarkson recently introduced the o-smoothed distance do(p,q) = 2d(p,q)/(d(o, p)+d(o,q)+d(p,q)) (where d denotes the Euclidean distance in the plane) as a geometric analogue of the Jaccard distance; its Voronoi diagrams can be used to determine for a query point q the site p maximizing the dilation (d(p,o)+d(o,q))/d(p,q) of p and q in a star network centered at o. Although smoothed distance is neither translation-invariant nor convex, we show how to transform these diagrams into minimization diagrams of translates of a convex function that takes the form f (x,y) = g(x)+h(y). For convex functions of this form, when g and h do not grow too quickly (as embodied in the constraints g′′′g′ < (g′′)2 and h′′′h′ < (h′′)2), the level sets of f form a family of pseudocircles in the plane, the minimization diagram of translates of f has linear complexity and can be constructed in O(n logn) randomized expected time, all cells in the minimization diagram are connected, and the set of bisectors separating any one cell in the diagram from each of the others forms an arrangement of pseudolines in the plane. Therefore, for sufficiently closely spaced points in the plane, the Voronoi diagram of smoothed distance has linear complexity and can be computed efficiently. We also experiment with using a variant of Lloyd’s algorithm, adapted to smoothed distance, to find uniformly spaced point samples with exponentially decreasing density around a point o.