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Given a set  of  moving points in fixed dimension , where the trajectory of each point is a polynomial of degree bounded by some constant, we present a kinetic data structure (KDS) for maintenance of the closest pair on . Assuming the closest pair distance is between 1 and  over time, our KDS uses  space and processes  events, each in worst-case time . Here,  is an extremely slow-growing function. The locality of the KDS is . Our closest pair KDS supports insertions and deletions of points. An insertion or deletion takes  time . Also, we use a similar approach to provide a KDS for the all -nearest neighbors in . The complexities of the previous KDSs, for both closest pair and all -nearest neighbors, have polylogarithmic factors, where the number of s depends on dimension . Assuming  is polynomial in , our KDSs obtain improvements on the previous KDSs. Our solutions are based on a kinetic clustering on . Though we use ideas from the previous clustering KDS by Hershberger, we simplify and improve his work.

A Clustering-Based Approach to Kinetic Closest Pair