A simple on-line randomized incremental algorithm for computing higher order Voronoi diagrams

We present a very simple algorithm for maintaining order-k Voronoi diagrams in the plane. By using a duatity transform that is of interest in its own right we show that the insertion or deletion of a site involves little more than the construction of a single convex hull in three-space. In particular, the order-k Voronoi dlagratn for n sites can becomputedin time 0(nk2 log n+nk log3 n) and optimal space O(k (n — k) ) by an on-line randomized incremental algorithm whose practictdity can be compared with the recent Voronoi dlagmrn algorithm by Guibas, Knrrth, and Sharir. The time bound can be improved by a log-factor without losing much sitnpticity. For k ~ log2 n, this is optimal for a randomized incremental construction; we show that the expected number of structural changes during the construction is t3(nk2 ). Finally, by going back to the primal space, we automatically obtain a dynamic data structure that supports k-nearest neighbor queries, insertions, and deletions in a planar set of sites. The structure is very easy to irnplemen~ exhibits a satisfactory expected performance, and occupies not more storage than the current order-k Voronoi diagram.

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