Enumerating the k closest pairs mechanically

Let $S$ be a set of $n$ points in $D$-dimensional space, where $D$ is a constant, and let $k$ be an integer between $1$ and $n \choose 2$. An algorithm is given that computes the $k$ closest pairs in the set $S$ in $O(n \log n + k)$ time, using $O(n+k)$ space. The algorithm fits in the algebraic decision tree model and is, therefore, optimal

An optimal algorithm for the on-line closest-pair problem

We give an algorithm that computes the closest pair in a set of $n$ points in $k$-dimensional space on-line, in $O(n \log n)$ ime. The algorithm only uses algebraic functions and, therefore, is optimal. The algorithm maintains a hierarchical subdivision of $k$-space into hyperrectangles, which is stored in a binary tree. Centroids are used to maintain a balanced decomposition of this tree

The largest hyper-rectangle in a three dimensional orthogonal polyhedron

Given a three dimensional orthogonal polyhedron P, we present a simple and efficient algorithm for finding the three dimensional orthogonal hyper-rectangle R of maximum volume, such that R is completely contained in P. Our algorithm finds out the three dimensional hyper-rectangle of maximum volume by using space sweep technique and enumerating all possible such rectangles. The presented algorithm runs in O(($n^2$+K)logn) time using O(n) space, where n is the number of vertices of the given polyhedron P and K is the number of reported three dimensional orthogonal hyper-rectangles for a problem instance, which is O($n^3$) in the worst case

Randomized Data Structures for the Dynamic Closest-Pair Problem

We describe a new randomized data structure, the {\em sparse partition}, for solving the dynamic closest-pair problem. Using this data structure the closest pair of a set of $n$ points in $k$-dimensional space, for any fixed $k$, can be found in constant time. If the points are chosen from a finite universe, and if the floor function is available at unit-cost, then the data structure supports insertions into and deletions from the set in expected $O(\log n)$ time and requires expected $O(n)$ space. Here, it is assumed that the updates are chosen by an adversary who does not know the random choices made by the data structure. The data structure can be modified to run in $O(\log^2 n)$ expected time per update in the algebraic decision tree model of computation. Even this version is more efficient than the currently best known deterministic algorithms for solving the problem for $k>1$