Extreme Distances in Multicolored Point Sets

Extreme Distances in Multicolored Point Sets

Given a set of n colored points in some d-dimensional Euclidean space, a bichromatic closest (resp. farthest) pair is a closest (resp. farthest) pair of points of different colors. We present efficient algorithms to maintain both a bichromatic closest pair and a bichromatic farthest pair when the the points are fixed but they dynamically change color. We do this by solving the more general prob...

On Point Sets with Many Unit Distances in Few Directions

We study the problem of the maximum number of unit distances among n points in the plane under the additional restriction that we count only those unit distances that occur in a xed set of k directions taking the maximum over all sets of n points and all sets of k directions We prove that for xed k and su ciently large n n k the extremal sets are essentially sections of lattices bounded by edge...

Similarity and Affine Invariant Distances Between 2D Point Sets

SERIE B | INFORMATIK Point Sets with Distinct Distances

For positive integers d and n let fd n denote the maximum cardinality of a subset of the n grid f ng with distinct mutual euclidean distances Improving earlier results of Erd os and Guy it will be shown that f n c n and for d that fd n cd n lnn where c cd are constants Also improvements of lower bounds of Erd os and Alon on the size of Sidon sets in f n g are given Furthermore it will be proven...

Faster Algorithms for Computing Distances between One-Dimensional Point Sets

Let S and T be two finite sets of points on the real line with |S| + |T | = n and |S| > |T |. We consider two distance measures between S and T that have applications in music information retrieval and computational biology: the surjection distance and the link distance. The former is called the restriction scaffold assignment problem in computational biology, and assigns each point of S to a p...