Location problems on networks have 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, where p is an arbitrary positive integer. The objective of the p-center problem is to locate p servers to minimize the maximum service cost among all clients, while the objective of the p-median problem is to minimize the sum of the service costs for all clients.
Traditionally, the service cost of a client is the weighted distance between the client and its nearest server. However, in some circumstances, the service may require a server to load or unload some package at some depot. Drezner and Wesolowsky proposed a generalization. In the generation, a set of depots X is given, and the service distance of a client is the distance from a server, to the client, to a depot, and back to the server, where the depot and the server are selected such that the total distance is minimized. The service cost of a client is defined as its weight multiplying the service distance of the client. This model is called the round-trip model. Later, Tamir and Halman considered an extended version, where each client ci is associated with a subset Xi □ X of depots that ci is allowed to use. This version is called the restricted version, and the original version is called the unrestricted version. Besides, they also proposed two extended service models: the customer one-way model and the depot one-way model, where the customer one-way model ignores the distance between the client and the server, and the depot one-way model ignores the distance between the depot and the server.
Tamir and Halman solved the restricted version of the round-trip and customer one-way 1-center problems on general graphs in O(mnlog n + m□i|Xi|) time, where n is the number of vertices and m is the number of edges. In this thesis, we improve their algorithms to O(mnlog n) time. When considering the unweighted case subject to X □ V, where V is the set of vertices and the unweighted case is the case that all the clients have the same weight, the running time is further reduced to O(mnloglog n + n2 log n) for the round-trip model and to O(mn + n2 log n) for the customer one-way model. Our technique can also be applied to the restricted version of the round-trip and customer one-way 1-median problems, and improves their running time.

為各種設施找出最佳的放置地點是一個日常生活中實際而且有著廣泛應用的問題，從交通運輸到網路通訊方面的設施設置問題都屬於這個領域，因此長久以來一直廣泛受到學者們的討論與研究。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 問題的定義是要放置 p 個伺服器，希望在服務所有使用者的花費中最貴的那一個能夠越便宜越好，而 p-median 問題則是要求服務所有使用者之花費的總和必須越小越好。
傳統上都將服務某一個使用者的花費定義為使用者到其最近伺服器的加權 (weighted) 距離，但某些設施的服務方式並不只是單純將服務送到使用者端而已，有時候還會需要在某些轉運站 (depot) 拿取物品或卸下貨物。Drezner 跟 Wesolowsky 於是提出了環形路徑 (round-trip) 模型。在這模型中，有一個轉運站集合稱為 X ，使用者可以使用任何一個 X 中的轉運站。他們將某一個使用者的服務距離定義為從伺服器到轉運站到使用者再回到伺服器的路徑長(注意他們會選擇使用能讓服務距離最短的伺服器以及轉運站)。而服務某一個使用者的花費就是加權的服務距離。之後，Tamir 跟 Halman 對這個問題提出一個延伸的版本。他們限制服務使用者 ci 時只能使用某些轉運站(這些轉運站的集合稱為 Xi )。他們將這個版本稱為限制版本 (restricted version)，並將原問題稱為無限制版本 (unrestricted version)。另外他們還提出了兩個延伸的服務模型，稱為使用者單向路徑 (customer one-way) 模型以及轉運站單向路徑 (depot one-way) 模型，其中使用者單向路徑模型不計算使用者到伺服器之間的距離，而轉運站單向路徑模型不計算轉運站到伺服器之間的距離。
本篇論文所探討的就是在一個廣義圖 (general graph) G = (V, E) 中尋找環形路徑模型以及使用者單向路徑模型下的最佳服務設施設置地點。本論文改進了在環形路徑以及使用者單向路徑模型下尋找一個中心 (center) 的演算法，將原本 O(mnlog n + m□i|Xi|) 時間的演算法加速到 O(mnlog n) 的時間，其中 m = |E| 而 n = |V|。若考慮不加權 (unweighted) 的情況並限制轉運站都在節點上，這個演算法可以再加速到 O(mnloglog n + n2 log n) 的時間。上述演算法所使用的技巧同樣可以套用在環形路徑以及使用者單向路徑模型下尋找一個重心 (median) 的問題，並且改進之前演算法的時間。

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