On Fully Diverse Sets of Geometric Objects and Graphs

Abstract

Diversity is a property of sets that shows how varied or different its elements are. We define full diversity in a metric space and study the maximum size of fully diverse sets. A set is fully diverse if each pair of elements is as distant as the maximum possible distance between any pair, up to a constant factor. We study metric spaces based on geometry, embeddings of graphs, and graphs themselves. In the geometric cases, we study measures like Hausdorff distance, Frechét distance, and area of symmetric difference between objects in a bounded region. In the embedding cases, we study planar embeddings of trees and planar graphs, and use the number of swaps in the rotation system as the metric. In the graph cases, we use the number of insertions and deletions of leaves or edges as the metric. In most cases, we show (almost) tight lower and upper bounds on the maximum size of fully diverse sets. Our results lead to a very simple randomized algorithm to generate large fully diverse sets in several cases.