Incremental Rank Updates for Moving Query Points

The query for retrieving the rank of all neighbors of a moving object at any given time, a continuous rank query, is an important case of continuous nearest neighbor (CNN) queries. An application for ranking queries is given by an ambulance driver who needs to keep track of the closest hospitals at all times. We present a set of incremental algorithms that facilitate efficient rank updates for some or all neighbors of a moving query point. The proposed algorithms allow us not only to maintain the exact rank of all n neighbors at any given time but also to track the rank of a subset of all neighbors. We show that updates for these continuous rank queries can be performed in linear time for arbitrary polygonal curves in two dimensions and in logarithmic time for movements along a fixed direction. Instead of using Voronoi diagrams, our algorithms are based on small subsets of all bisectors between neighbors. We prove that it is sufficient to keep track of only n− 1 bisectors for all n neighbors. The algorithms for maintaining the rank only require minimal incremental updates on the bisector sets.