Kinetic and dynamic data structures for closest pair and all nearest neighbors

We present simple, fully dynamic and kinetic data structures, which are variants of a dynamic two-dimensional range tree, for maintaining the closest pair and all nearest neighbors for a set of n moving points in the plane; insertions and deletions of points are also allowed. If no insertions or deletions take place, the structure for the closest pair uses O(n log n) space, and processes O(n²Β_s+2(n)log n) critical events, each in O(log²n) time. Here s is the maximum number of times where the distances between any two specific pairs of points can become equal, Β_s(q) = _s(q)/q, and _s(q) is the maximum length of Davenport-Schinzel sequences of order s on q symbols. The dynamic version of the problem incurs a slight degradation in performance: If m n insertions and deletions are performed, the structure still uses O(n log n) space, and processes O(mnΒ_s+2(n)log³ n) events, each in O(log³n) time. Our kinetic data structure for all nearest neighbors uses O(n log² n) space, and processes O(n ²Β²_s+2(n)log³ n) critical events. The expected time to process all events is O(n²Β _s+2²(n) log⁴n), though processing a single event may take Θ(n) expected time in the worst case. If m n insertions and deletions are performed, then the expected number of events is O(mnΒ²_s+2(n) log³n) and processing them all takes O(mnΒ²_s+2(n) log⁴n). An insertion or deletion takes O(n) expected time.