The Colored Sector Search Tree: A Dynamic Data Structure for Efficient High Dimensional Nearest-Foreign-Neighbor Queries

Abstract

In this paper we present the new data structure Colored Sector Search Tree (CSST) for solving the Nearest-Foreign-Neighbor Query Problem (NFNQP): Given a set S of n colored points in ℝ^D, where D ≥ 2 is a constant, and a subset S′ ⊂ S stored in a CSST, for any colored query point q ∈ ℝ^D a nearest foreign neighbor in S′, i.e. a closest point with a different color, can be reported in O(log n(log log n)D ^{D − 1}) time w.r.t. a polyhedral distance function that is defined by a star-shaped polyhedron with O(1) vertices; note that this includes the Minkowski metrics d ₁ and d _∞. It takes a preprocessing time of O(n(log n)D ^{D − 1}) to construct the CSST. Points from S can be inserted into the set S′ and removed from S′ in O(log n(log log n)D ^{D − 1}) time. The CSST uses O(n(log n)D ^{D − 1}) space. We present an application of the data structure in the parallel simulation of solute transport in aquifer systems by particle tracking. Other applications may be found in GIS (geo information systems) and in CAD (computer aided design). To our knowledge the CSST is the first data structure to be reported for the NFNQP.