本篇論文主要在探討directed Stenier tree problem(DSTP)的近似演算法。Directed Steiner Tree問題為：給定一有向加權圖 ， 為節點集合， 為邊的集合，對於 中的任一條邊上都存在一個實數 ，稱 為此邊的成本。此外， 集合中存在一個節點 以及一個子集合 ，稱此節點 為root， 集合為terminal集合。目標為在 中找到一棵以 為root，衍生至 集合中所有節點的最小成本樹，也就是說，此最小成本樹包含 至 集合中所有節點的路徑，所有節點均無任何限制。其中我們研究以Charikar et al.等人所提出的DSTP演算法為基準，它能在 的時間內得到approximation bound為 的近似解， 為節點數目， 為terminal數目， 為演算法的遞迴層數。而T. W. Chang則是將Charikar et al.等人所提出的DSTP演算法中過多的重複運算給精簡、並改良出演算法modified Charikar’s algorithm with improvement on terminal(MCAIT)，可大幅提昇執行效率。
本篇論文基於Charikar及T. W. Chang的演算法，加入了degree-constrained的觀念，給予節點分枝限制，推導出此種狀況下l-restricted tree的bound並設計出針對以上演算法的degree-constrained對應版本。由模擬結果可看出degree-constrained steinr tree problem與DSTP演算法在時間上與成本上的關聯性。

This thesis focused on discussing the approximation algorithm for Directed Stenier Tree Problem (DSTP). The following is the definition of the DSTP: given a directed weighted graph , which is the node set; is the collection of edges . There is a real number on every edge in graph , which represents the cost for the edge. Moreover, a node and a sub-set exist in the node set. The node is defined as the Root and collection of is the Terminal Set. The objective is to find the tree with minimum cost in graph G; with root on r and expands to other nodes in the X sub-set. That is, the tree with minimum cost contains all the edges from r to X sub-set, and there is no restriction for any node.
The research team study the DSTP algorithm propounded by Charikar et al. and other members. Based on DSTP Algorithm, we can get a approximation ratio with within time of , where n is the number of nodes; k is the number of terminals; is the recursion of the algorithm.
Based on DSTP Algorithm propounded by Charikar et al., T. W. Chang shorten the calculation process and propounded an advanced algorithm named Modified Charikar’s Algorithm with Improvement on Terminal (MCAIT). With the MCAIT, we can highly improve the efficiency when solving the problem.
This thesis was based on the algorithms propounded by both Charikar and T. W. Chang. By adding the concept of degree-constrained, as well as other restrictions to the node, we derived the bound for the l-restricted tree under this circumstance. The relationship of time and cost between degree-constrained Steiner Tree problem and DSTP Algorithm were shown in the simulation.

摘要 I
目錄 II
圖目錄 IV
表格目錄 V
第一章 簡介 1
第二章 相關演算法 3
2.1. Basic Definition 3
2.2. l-restricted tree 4
2.3. Charikar’s algorithm 5
2.4. MCAIT 9
2.4.1. Charikar’s algorithm問題分析 9
2.4.2. 至 中過多的重複運算之改進概念 10
第三章 Degree-constrained Directed Steiner Tree Problem 12
3.1. Degree-constrained 問題分析 12
3.2. DCCA 12
3.3. DCCAIT 20
3.4. Approximation ratio 30
3.5. 時間複雜度 31
3.5.1. Charikar’s algorithm時間複雜度分析 32
3.5.2. DCCA時間複雜度分析 32
3.5.3. DCCAIT時間複雜度分析 33
3.5.4. 各演算法時間複雜度比較 34
第四章 模擬與結果 35
4.1. 環境設定 35
4.2. 模擬結果與分析 35
4.2.1. Randomly fan-out degree 36
4.2.2. Sparse splitting nodes 38
第五章 結論 42
參考文獻 43

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