An optimal parallel algorithm for the all-nearest-foreign-neighbors problem in arbitrary dimensions

Given a set S of n points in R^D, D⩾2. Each point p∈S is assigned a color c(p) chosen from a fixed color set. The All-Nearest-Foreign-Neighbors Problem (ANFNP) is to find for each point p∈S its nearest foreign neighbors, i.e. the set of all points in S/{p} that are closest to p among the points in S with a color different from c(p). We introduce the Well Separated Color Decomposition (WSCD) which gives an optimal O(log n) parallel algorithm to solve the AMFNP, for fixed dimension D⩾2 and fixed L^t-metric d_t, 1⩽t⩽∞. The WSCD is based upon the Well Separated Pair Decomposition (Callahan et al., 1992). The ANFNP finds extensive applications in VLSI design and verification for two dimensions, and in traffic-control systems and Geographic Information Systems for D>2 dimensions. To the best of our knowledge, this is the only known optimal parallel algorithm for the ANFNP