Limit Theorems for the Cost of Splitting Random Cayley Trees

• Main Authors: Kuo, Yen-Ting, 郭彥廷
• Other Authors: Michael Fuchs
• Format: Others
• Language: en_US
• Published: 2012
• Online Access: http://ndltd.ncl.edu.tw/handle/24068832006181372227

碩士 === 國立交通大學 === 應用數學系所 === 100 === In computer science, the so-called "Union-Find problem" is concerned with establishing a data structure for maintaining a collection of disjoint sets such that the process of merging sets can be carried out efficiently. Indeed, several data structures and corresponding algorithms for merging sets have been proposed. For the purpose of comparing the complexity of these algorithms, it is naturally to consider the total cost incurred from merging n singleton sets into one set. In this thesis, we assume that the cost of each merging step is the power of the sum of the sizes of the sets being merged and then derive the expected value and the limiting distribution of the total cost under the random spanning tree model. The main tool used in this thesis is singularity analysis, which is a method connecting the asymptotics of generating functions with the asymptotics of their coefficients. We will use it to derive the moments of each order. Then, with the method of moments, the limiting distribution of the total cost will follow. In the first part of this thesis (Chapter 1), we introduce the problem of interest and state the results of our work. In Chapter 2, we give an introduction about our main tool, singularity analysis. The central part of this thesis, namely Chapter 3, is devoted to the derivation of the expected value and higher moments. These results will will then be used in Chapter 4 for prove our main result. Finally, we end the thesis with a conclusion in Chapter 5.