Optimal Networks

TL;DR

This paper introduces the theory of optimal networks, focusing on connecting points efficiently in metric spaces through spanning trees, shortest trees, and minimal fillings, with applications in computational geometry.

Contribution

It provides an introductory overview of optimal networks, detailing three key types: spanning trees, shortest trees, and minimal fillings, with insights into their properties and applications.

Findings

Analysis of spanning trees and their optimality

Characterization of shortest trees and locally shortest trees

Introduction to minimal fillings and their significance

Abstract

This mini-course was given in the First Yaroslavl Summer School on Discrete and Computational Geometry in August 2012, organized by International Delaunay Laboratory "Discrete and Computational Geometry" of Demidov Yaroslavl State University. The aim of this mini-course is to give an introduction in Optimal Networks theory. Optimal networks appear as solutions of the following natural problem: How to connect a finite set of points in a metric space in an optimal way? We cover three most natural types of optimal connection: spanning trees connection without additional road forks, shortest trees and locally shortest trees, and minimal fillings.