Nearest neighbors search using point location in balls with applications to approximate Voronoi decompositions

We present alternate reductions of the nearest neighbor searching problem to Point Location in Balls that reduces the space bound of Sariel Har-Peled's [S. Har-Peled, A replacement for Voronoi diagrams of near linear size, in: Proc. of IEEE FOCS, 2001, pp. 94–103, full version available from http://www.uiuc.edu/~sariel/papers] recent result on Approximate Voronoi Diagrams to linear while maintaining the logarithmic search time. We do this by simplifying the construction of [S. Har-Peled, A replacement for Voronoi diagrams of near linear size, in: Proc. of IEEE FOCS, 2001, pp. 94–103, full version available from http://www.uiuc.edu/~sariel/papers] that reduces the number of balls generated by algorithm by a logarithmic factor to O(nlogn). We further reduce the number of balls by a new hierarchical decomposition scheme and a generalization of PLEBs to achieve linear space decomposition for nearest neighbor searching. The construction of our data structures takes O(nlogn) time.