網絡放置理論 (network location theory) 中最常研究的問題是p中心 (p-center) 和p重心 (p-median) 問題，在這些問題中需要選擇p個頂點。p 中心的目標函數是從每個頂點到所選擇的p個頂點之間的最大加權距離。p重心的目標函數是所有頂點到所選擇的p個頂點之間的加權距離之和。p中重心 (p-centdian) 問題同時考慮中心函數和重心函數。在本論文中，我們研究區塊圖 (block graph) 上的連通p中重心問題，這個問題要求所選擇的p個頂點的導出子圖是連通的。我們考慮了連通p中重心問題的三個變體。第一個稱為帕雷托最優 (Pareto-optimal) 連通p中重心問題 (P-CpCD)，其目標是找到中心函數和重心函數的所有帕雷托最優的解集合；第二個稱為連通p中重心問題 (CpCD)，其目標是找到一個最佳解，可以最小化重心函數和中心函數按給定係數 (λ, 1  λ) 的凸組合，其中0 ≤ λ ≤ 1；第三個稱為帶有動態係數的連接 p中重心問題 (Dynamic-CpCD)，其目標是建構一個針對輸入圖形的資料結構；之後對任意一個接收到的查詢參數 λ，其中 0 ≤ λ ≤ 1，都可以快速找到一個最佳解，可以最小化重心函數和中心函數按給定係數 (λ, 1  λ) 的凸組合。
對於 P-CpCD，Kang et al. 提出了一個 O(n2) 時間的演算法。我們首先提供一個 O(np) 時間的改進演算法。接著，我們提供一個 O(n) 時間的演算法，這個演算法會以壓縮的格式輸出解答。此外，我們提出了一種新的區塊圖表示方法，並展示了當區塊圖以此表示法輸入時，我們的演算法運行時間為 O(b)，其中 b 是輸入圖中區塊的數量。對於 CpCD 問題，在 λ ∈ [1/2, 1] 的請況下，Nguyen et al. 提出了一個 O(n) 時間的演算法，並且將λ ∈ (0, 1/2) 時，是否能在O(n) 時間解決該問題留作一個未解決的問題。我們提出一個對於任意 λ ∈ [0, 1] 都適用的 O(n) 時間算法來解決這個問題。對於dynamic-CpCD，我們提供了一個演算法，在經過 O(n) 時間的預處理後，每個查詢可以在 O(log n + p) 時間內得到回答。

The most common problems studied in network location theory are the p-center and the p-median problems, in which p vertices are to be selected. The p-center objective function is the maximum weighted distance from each vertex to the selected p vertices. The p-median objective function is the sum of weighted distances from all vertices to the selected p vertices. The p-centdian problem is concerned with both the center and the median function at the same time. In this thesis, we study the connected p-centdian problem on block graphs, in which the induced subgraph of the selected p vertices is asked to be connected. Three variants of the connected p-centdian problem are considered. The first is called the Pareto-optimal connected p-centdian problem (P-CpCD), which is to find all Pareto-optimal p-vertex sets of the center and median functions; the second is called the connected p-centdian problem (CpCD), which is to find a set of connected p-vertex to minimize the convex combination of the median and the center functions with given coefficients (λ, 1  λ), where 0  λ  1; and the third is called the connected p-centdian problem with dynamic coefficients (dynamic-CpCD), which is to build for an input graph a data structure that receives a parameter λ, where 0  λ  1, and returns a connected p-vertex set that minimizes the convex combination of the median and the center functions with coefficients (λ, 1  λ).
For P-CpCD, Kang et al. had an O(n2)-time algorithm. First, we give an improved O(np)-time algorithm. Then, we present an O(n)-time algorithm, which outputs the solutions in a compact form. Furthermore, we propose a new representation of block graphs and show that our algorithm runs in O(b) time when the input block graph is given in this representation, where b is the number of blocks in the input graphs. For CpCD, Nguyen et al. had an O(n)-time algorithm for λ ∈ [1/2, 1] and left solving the problem in O(n) time for λ ∈ (0, 1/2) as an open problem. We answer this open problem by presenting an O(n)-time algorithm for any λ ∈ [0, 1]. For dynamic-CpCD, we provide an algorithm that answers each query in O(log n + p) time after an O(n)-time preprocessing.

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