An Optimal Algorithm for Closest-Pair Maintenance

Given a set S of n points in k-dimensional space, and an Lt metric, the dynamic closest-pair problem is defined as follows: find a closest pair of S after each update of S (the insertion or the deletion of a point). For fixed dimension k and fixed metric Lt, 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 the algorithm is optimal up to a constant factor because Ω(log n) is a lower bound, in an 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

Dynamic rectangular point location, with an application to the closest pair problem

In the $k$-dimensional rectangular point location problem, we have to store a set of $n$ non-overlapping axes-parallel hyperrectangles in a data structure, such that the following operations can be performed efficiently: point location queries, insertions and deletions of hyperrectangles, and splitting and merging of hyperrectangles. A linear size data structure is given for this problem, allowing queries to be solved in $O((\log n)^{k-1} \log\log n )$ time, and allowing the four update operations to be performed in $O((\log n)^{2} \log\log n)$ amortized time. If only queries, insertions and split operations have to be supported, the $\log\log n$ factors disappear. The data structure is based on the skewer tree of Edelsbrunner, Haring and Hilbert and uses dynamic fractional cascading. This result is used to obtain a linear size data structure that maintains the closest pair in a set of $n$ points in $k$-dimensional space, when points are inserted. This structure has an $O((\log n)^{k-1})$ amortized insertion time. This leads to an on-line algorithm for computing the closest pair in a point set in $O( n (\log n)^{k-1})$ time