A SIMPLE ON-LINE RANDOMIZED INCREMENTAL ALGORITHM FOR COMPUTING HIGHER ORDER VORONOI DIAGRAMS

Abstract

We present a simple algorithm for maintaining order-k Voronoi diagrams in the plane. By using a duality 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 diagram for n sites can be computed in time [Formula: see text] and optimal space [Formula: see text] by an on-line randomized incremental algorithm. The time bound can be improved by a logarithmic factor without losing much simplicity. For k≥ log 2 n, this is optimal for a randomized incremental construction; we show that the expected number of structural changes during the construction is ⊝(nk2). Finally, by going back to primal space, we obtain a dynamic data structure that supports k-nearest neighbor queries, insertions, and deletions in a planar set of sites. The structure promises easy implementation, exhibits a satisfactory expected performance, and occupies no more storage than the current order-k Voronoi diagram.