A study on some 2-clustering problems

• Main Authors: LIANG,QIU-GUO, 梁秋國
• Other Authors: LI,JIA-TONG
• Format: Others
• Language: zh-TW
• Published: 1990
• Online Access: http://ndltd.ncl.edu.tw/handle/60959783890198587699

博士 === 國立清華大學 === 資訊科學研究所 === 78 === Consider a set of communication posts on a large plane. Each post is equipped with a transmitter that can reach some distance t from the post. If all the posts are within distance t of each other, all can communicate without difficulty. Suppose, however, that some posts are further than t units apart. We would like to know if the posts can be split into two groups so that within each group each pair of posts may communicate. In order to conserve energy, we wish to reduce the power of the transmitters as much as possible. Accordingly, what is the smallest t for which such a partition into two group is possible﹖ The above problem is typical of a large class of problems that have been studied reccntly, that concern the clustering of a set of objects. The problem of clustering a set of objects arises in many disciplines, for example, data compression, pattern recognition and service site assignment. Because of the wide range of applications, there are many variations of this problem. The main difference between these problems is in the objective function. Many different objective functions have been considered by many researchers. These objective functions usually depend on the dissimilarity between any two objects. Generally speaking, the clustering problem is to partition a set of n objects into k nonempty disjoint subsets, called clusters. Many computational complexities have been discovered for different clustering problems. Since many clustering problems, when k>=3 have been proved to be NP- complete, we concentrate on the clustering problems when k=2, or 2- clustering problems. In this dissertation, we discuss the following 2- clustering problems: the dual satisfaction problem, the farthest pair partition problem, the constrained farthest pair partition problem, minimum diameter partition problem, the specified diameter partition problem and the Euclidean mini- sum 2- clustering problem. The first three problems are defined and solved on the graph model. We propose a uniform approach, called the spanning tree vertex labeling approach, to partition the input vertex set into two disjoint subsets. Essentially, we construct a minimum, or maximum spanning tree, depending upon the problem. We then partition the vertices based upon the tree constructed. Our approach is easier to understand and easier to implement. The last three problems are discussed on Euclidean plane. that is, the input of the problem is a set of points in 2- dimensional plane. We discover some geometric properties that can be used to find the optimal solution efficiently. The geometric properties are developtd by using the concept of maximum spanning tree of points. We found that the reason that we can devise more better algorithms than others is we do not consider so many pairs of points. In fact, we only consider those pairs of points that related to the maximum spanning tree of points.