Diameter, width, closest line pair, and parametric searching

Time-Windowed Closest Pair

Given a set of points in any constant dimension, each of which is associated with a time during which that point is active, we design a data structure with O(n log n) space that can find the closest pair of active points within a query interval of time in O(log log n) time using a quadtree-based approach in the word-RAM model.

Complexity, the Changing Minimum and Closest Pair

1 Las Vegas and Monte Carlo algorithms Definition 1.1 A Las Vegas algorithm!Las Vegas algorithm is a randomized algorithms that always return the correct result. The only variant is that it's running time might change between executions. An example for a Las Vegas algorithm is the QuickSort algorithm. Definition 1.2 Monte Carlo algorithm!Monte Carlo algorithm is a randomized algorithm that migh...

Parametric Searching

An Optimal Algorithm for Closest-Pair Maintenance

Given a set S of n points in k-dimensional space, and an L t metric, the dynamic closest pair problem is deened 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 algo...

2D Closest Pair Problem: A Closer Look

A closer look is taken at the well-known divide-andconquer algorithm for finding the closest pair of a set of points in the plane under the Euclidean distance. An argument is made that it is sufficient, and sometimes necessary, to check only the next three points following the current point associated with the y-sorted array in the combine phase of the algorithm.