APPROXIMATE k-NEAREST NEIGHBOR GRAPH ON MOVING POINTS

In this paper, we introduce an approximation for the k-nearest neighbor graph (k-NNG) on a point set P in R^d. For any given ε 0, we construct a graph, that we call the approximate k-NNG, where the edge with the ith smallest length incident to a point p in this graph is within a factor of (1 + ε) of the length of the edge with the ith smallest length incident to p in the k-NNG. For a set P of n moving points in R^d, where the trajectory of each point p ∈ P is given by d polynomial functions of constant bounded degree, where each function gives one of the d coordinates of p, we compute the number of combinatorial changes to the approximate k-NNG, and provide a kinetic data structure (KDS) for maintenance of the edges of the approximate k-NNG over time. Our KDS processes O(kn² log^d+1 n) events, each in time O(log^d+1 n).