Exact Solutions of Shortest-Path Problems Based on Mechanical Analogies: in connection with labyrinths, mazes and graph theory

In this book, approaches based on mechanical analogies are presented for the solutions of path finding problems and exact solutions of shortest path problems. Shortest path problems are of great importance not only in terms of theory but also in solutions of optimization problems in many different areas of real life. The fact that shortest path problems are spread over different areas makes it important that it is understandable, even to a certain level, by people of different branches and education levels in order to use the proposed solution methods effectively. In the preparation of this book, special attention was paid to this issue, and the familiar nature of mechanical behaviors was supported by visuals that could be easily understood by everyone, and the theory of the essence of the approach was made without allowing it to be lost due to detailed presentations of numerical methods that are already well known. The numerical methods in the book are utilized in the programs commonly used in calculations and simulations of the engineering and the gaming industry. Faster progress can be made in multidisciplinary working groups on the adaptation of the finite element method (FEM) based programs or rigid body dynamics (RBD) based motion engines to presented approaches. In this book, not even an equation was required to present topics and approaches. Because once the fiction of mechanical behaviors is designed with a natural imagination, the only thing left for the solution of the problem is the introduction of the designed model into software created on the basis of well-known numerical methods. In the study, the terms maze and labyrinth are frequently used. Although these two terms historically refer to some geometric forms, Graph Theory and topology also express certain definitions. It is important to understand the “labyrinth-path finding” and “maze-shortest path” relationship, especially for those who will use the methods to be presented with their engineering approach, in connection with these broadly detailed definitions in the study. This book is organized into four chapters. The articles in each chapter are prepared independently of each other. Although the articles are independent from each other, since the approach in each chapter covers the approach in the previous chapter, reading articles in order facilitates their understanding. In Chapter 1 and 2, each path finding problem is addressed with different mechanical analogies, and there are important differences between approaches in terms of both computational cost and criteria used in the solutions. Chapter 3 provides highly detailed information and linked solutions for situations that need attention when it comes to implementing mechanical modeling and numerical methods. In Chapter 4, a very effective and simplified method based on the displacement criteria that can be used in the exact solution of the shortest path problems constructed in the light of the warnings mentioned in Chapter 3 is presented. FEM, which engineers and scientists are quite familiar with, has been widely used in presenting approaches and simulations, but RBD-based calculations also have significant advantages such as computational cost. The main reason for the predominant use of FEM as a numerical method in the examples is the fact that FEM has many parameters that allow it to be adapted to different problem types easily and is more effective in understanding the approaches. The topics in the book are quite different from my routine academic work, and the writing of the book has been a long process due to ongoing projects, studies and contributions to education. The covid19 pandemic provided the time for me to finish this book. I hope this book will contribute to the work of researchers interested in the subject and serve as an additional toolbox that can be used in the exact solution of shortest problems.