Given a set S of n points in k-dimensional space, and an L t metric, the dynamic closest pair problem is de ned as follows: nd a closest pair of S after each update of S (the insertion or the deletion of a point). For xed dimension k and xed metric L t , we give a data structure of size O(n) that maintains a closest pair of S in O(log n) time per insertion and deletion. The running time of algorithm is optimal up to constant factor because (log n) is a lower bound, in algebraic decision-tree model of computation, on the time complexity of any algorithm that maintains the closest pair (for k = 1). The algorithm is based on the fair-split tree. The constant factor in the update time is exponential in the dimension. We modify the fair-split tree to reduce it.
In REF an optimal algorithm for maintaining the closest pair of points is described.
An Optimal Algorithm for Closest Pair Maintenance
Discrete Comput. Geom
Mathematics Computer Science