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

Abstract

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 ( n 2 β s +2 ( n )log n ) critical events, each in O (log 2 n ) 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 ) = λ s ( q )/ q , and λ 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 ( mn β s +2( n )log 3 n ) events, each in O (log 3 n ) time. Our kinetic data structure for all nearest neighbors uses O ( n log 2 n ) space, and processes O ( n 2 β 2 s +2 ( n )log 3 n ) critical events. The expected time to process all events is O ( n 2 β s +2 2 ( n ) log 4 n ), 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 ( mn β 2 s +2 ( n ) log 3 n ) and processing them all takes O ( mn β 2 s +2 ( n ) log 4 n ). An insertion or deletion takes O ( n ) expected time.