為裝置在網路中安置最佳的位置是一個日常生活中真實而且重要的問題。它被各領域的學者們廣泛的研究與討論，尤其是傳輸與通訊領域為最甚。諸多最佳化的條件當中以『距離和』與『離心率』兩者被引用的最為頻繁。距離和的意思是網路上所有點到裝置的距離加總，而離心率指的是網路當中離裝置最遠之點與裝置的距離。由於存在各種型態的裝置與不同的最佳化條件，因此有許許多多的配置問題被衍生出來。該類問題引人之處在於下列三點：一、該類問題是真實而重要的問題。二、到目前為止有許多該類的問題仍然沒有有效率的解決方法。三、有許多該類問題缺乏平行演算法的討論。大部分的配置問題在廣義的圖上都被證明是NP-Hard，比如說p重心問題、樹狀重心問題、及限制長度的路徑狀重心問題。因此，學者們則紛紛地將注意力專注在某一些特別的圖上。而其中，樹狀圖受到最多的青睞。在此文中也將專注於樹狀圖之研究。目前，我對這個主題研究之成果如下。首先，我改善了一些眾所週知的問題之計算速度。例如，限制長度的路徑狀重心問題(從O(n2)加快到O(nlog nloglog n))、限制長度的最重路徑問題(從O(nlog2 n)加快到O(nlog n))、及無權值之雙重心問題(從O(nlog n)加快到O(n))。其次，我證明了兩個在設計配置問題演算法時經常會遭遇到的問題之極限計算速度。第三，我定義了一個叫做f匹配的問題，並且為之設計線性時間的演算法。這個問題的解決能在設計諸多無權值配置問題演算法時有莫大的幫助。

Finding the optimal location of a facility on a network is a real and significant problem in our daily life. It has been widely investigated by researchers of various fields, especially in the fields of transportation and communication. The criteria for optimality discussed extensively in the literature are distance-sum and eccentricity. The distance-sum of a facility is the total distance from all vertices to it. The eccentricity of a facility is the distance from the farthest vertex to it. Due to the variety of facility kinds and different criteria for optimality, many location problems have been defined and studied. The reasons why we are interested in this kind of problems are as follows. 1) It is a real and important problem. 2) Many location problems have not been solved efficiently so far. 3) There are still many problems without efficient parallel algorithms. Most of location problems are NP-Hard on general graphs, such as the problems of locating p-median, locating tree-shaped median, and locating minimum distance-sum path of a specified length. Thus, investigators focus on some special classes of graphs. Trees get the most attention. In this thesis, all problems are discussed on trees. We have several the following results. First, we propose efficient algorithms to improve upper bounds for many well-known location problems. For examples, we improve the upper bound for the problem of finding a core of a specified length from O(n2) time to O(nlog nloglog n) time, the upper bound for the problem of finding a length-constrained heaviest path from O(nlog2 n) time to O(nlog n) time, and the upper bound for the problem of finding a 2-median on an unweighted tree from O(nlog n) time to O(n) time. Secondly, we provide W(nlog n) lower bounds for two problems that are encountered frequently in the designing of several location problems. Thirdly, we define a problem called the integral f-left match problem and provide it an O(n) time algorithm. The proposed algorithm can be applied to solve many location problems on unweighted trees in O(n) time.

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