| Summary: | 碩士 === 國立清華大學 === 資訊工程學系 === 90 === Location theory on networks has been widely investigated by researchers from different fields for more than thirty years due to their significance and practical value. Among various location problems, the p-center and the p-median problems are the most common. The objective of the p-center problem is to locate p service centers on a network with n nodes to minimize the maximum of the service distances of the clients to their respective nearest service center, while the objective of the p-median problem is to minimize the sum of these service distances. Usually, the term "single-facility" indicates the case of p=1, and "multi-facility" indicates the case of p>2. When p>2 is an arbitrary integer, the two problems on general networks are NP-hard. Therefore, most researchers have devoted to the case p=1 or the case that the networks under consideration are trees.
Slater introduced a generalization of the above two problems, which is called the p-facility k-centrum problem. The objective is to minimize the sum of the k largest service distances. When k=1, the problem is just the p-center problem; when k=n, the problem becomes the p-median problem. Therefore, the generalization unifies the two essential problems. Due to the NP-hardness, previous studies on the k-centrum problem also focused on the case p=1 or the case that the networks under consideration are trees.
The single-facility k-centrum problem on a tree is the focus of this thesis. For this problem, Tamir had an O(nlog2 n) time algorithm. In this thesis, an improved algorithm is proposed. The proposed algorithm requires O(nlog n) time.
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