Location theory這類問題有著相當實際與廣泛的應用，從交通運輸到網路傳輸方面的設施設置問題都屬於這個領域，因此長久以來一直受到許多學者的注意與討論。在Location theory 上最具有代表性也最重要的兩個問題，是 p中心（p-center）問題以及 p 重心（p-median）問題。p-center問題的定義是要在有 n 個節點的網路上放置 p 個伺服器，希望在所有使用者的服務距離（即每一個使用者到最靠近自己之伺服器的距離）中最遠的那一段能夠越短越好，而p-median問題則是要求所有使用者之服務距離的總和必須越小越好。在分類上來說， p=1 時稱之為單設施 (single-facility) 問題，而當 p>1 時，則被稱之為多設施 (multi-facility) 問題。由於在 p>2 時，這兩個問題在一般的網路架構下都是NP-hard，因此相關的研究都集中在 p=1 或樹狀網路架構下。
Slater 提出了一個涵括 p-center 與 p-median 這兩個問題的廣義問題，稱之為 p-facility k-centrum問題。這個問題的定義是要網路上設置 p 個設施，讓所有服務距離中最遠的前k段距離之總和為最小。這個問題在k=1的情形下就是 p-center 問題，而在k=n的情形下，就會成為 p-median 問題。由於這個問題在一般的網路架構下也一樣是NP-hard，所以過去的相關討論同樣地集中在 p=1 或樹狀網路架構下。

這篇論文所探討的主題，是在樹狀網路上的單設施 k-centrun 問題。Tamir曾經針對這個問題設計了一個O(nlog2 n) time 的演算法。而這篇論文將會先探討有關單設施 k-centrum 的性質，然後再提出一個 O(nlog n) time 的演算法。

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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