Graph and Geometric Algorithms and Efficient Data Structures

Abstract

Many NP-complete optimization problems may be approximately solved by stochastic or deterministic heuristic methods and it is necessary to find their efficient data representation to minimize iteration computational time. In this chapter, we will touch the Minimum Steiner Tree Problems in Graphs (or Network Steiner Tree Problem), which can be solved by heuristics based on the Minimum Spanning Tree Problem and/or the Shortest Path Problem using a binary heap that enables to implement a priority queue that substantially increases the algorithm efficiency. We will also show a Delaunay triangulation-based way of finding minimal networks connecting a set of given points in the Euclidean plane using straight lines (minimum spanning tree) and its more general case (Steiner minimum tree) where additional points can be considered. Finally, we will deal with visibility graphs, Voronoi diagrams and rapidly exploring trees and focus on their applications in robot motion planning, where the robot should pass around obstacles from a given starting position to a given target position, touching none of them.