[Hungaraian Method]

by

ACKNOWLEDGEMENT

I would take this opportunity to thank my research supervisor, family and friends for their support and guidance without which this research would not have been possible.

DECLARATION

I, [type your full first names and surname here], declare that the contents of this dissertation/thesis represent my own unaided work, and that the dissertation/thesis has not previously been submitted for academic examination towards any qualification. Furthermore, it represents my own opinions and not necessarily those of the University.

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ABSTRACT

We propose a novel hierarchical clustering algorithm for data-sets in which only pairwise distances between the points are provided. The classical Hungarian method is an efficient algorithm for solving the problem of minimal-weight cycle cover. We utilize the Hungarian method as the basic building block of our clustering algorithm. The disjoint cycles, produced by the Hungarian method, are viewed as a partition of the data-set. The clustering algorithm is formed by hierarchical merging. The proposed algorithm can handle data that is arranged in non-convex sets. The number of the clusters is automatically found as part of the clustering process. We report an improved performance of our algorithm in a variety of examples and compare it to the spectral clustering algorithm.

TABLE OF CONTENTS

ACKNOWLEDGEMENTII

DECLARATIONIII

ABSTRACTIV

CHAPTER 1: INTRODUCTION6

Problem Background6

CHAPTER 2: LITERATURE REVIEW8

Direct Kuhn Algorithm8

Kuhn Algorithm with Contour Integral8

Numerical Examples9

Numerical Solutions to Eigenvalue Equations10

Analysis of Kuhn (Hungarian Algorithm)10

Combinatorial optimization11

Modeling combinatorial optimization problems into multi-entity systems12

Local fitness function13

Combinatorial Approaches14

The Parameter Space16

Totally stochastic methods of library preparation and modification18

The Thin Film Deposition Based Method18

Numerical Simulations20

Fragmentation Approaches and Fragment Space21

Fragment Frequency Analysis and Fragment Mining23

Fragment Tree25

Fragment Docking And Virtual Fragment Screening28

Strategies to Assemble and Optimize Fragments34

Fragment-Based Growing and Linking Strategies35

Combinatorics37

Combinations and Permutations38

Graph Theory41

Combinatorial Designs41

Asymptotics44

Connections with Computer Science45

Concluding Remarks46

CHAPTER 3: METHODOLOGY47

REFERENCES49

CHAPTER 1: INTRODUCTION

The numerical solutions to complex polynomial and transcendental equations are problems that are often met in the fields of electromagnetic theory and engineering. For example, in the Prony method for processing transient data and the simplified real frequency technique used to design broad-band microstrip antennas, the root finding of the polynomial is needed. At the same time, for the problems associated with the computation of the natural frequency in the singularity expansion method (SEM), the extraction of the dielectric parameters of materials from the measurement data, the evaluation of the dispersion relation of a microstrip line and the propagation constant of a leaky-wave antenna, and so forth, the tough task of solving a complex transcendental equation is always involved.

Mostly iterative methods are used to resolve those equations. However, the iterative method needs the initial values and the selection of those is rather difficult. In this article the Kuhn algorithm, which is designed to obtain all the roots of a polynomial equation, is introduced for the first time. This algorithm is more convenient than the iterative method in solving polynomial equations because no initial values are required and the derivative of the equation is not needed. It also has been proved that the Kuhn algorithm is much more efficient than the Newton iterative method for polynomial equations (Goldberg, 1989, ...