Ku 等作者在GLOBECOM 2012首先提出在信賴模型下的檔案傳輸問題。
他們把信賴關係定義成一棵完全二元樹並且設計了一個使用最少回數的方
法讓檔案從樹根傳至所有其他使用者。Tien 等作者(2016) 把信賴關係延
伸成有根的二元有向無環圖(DAG) 並且設計了較簡單的最佳回數演算法。
在本論文中,我們進一步把信賴關係延伸成一般的有向無環圖。我們證
明當有向無環圖的最大外分支度至少為6時,則找出使用最少回數的傳輸順
序將會是一個NP困難的問題。另一方面, 當信賴關係為一個一般樹,且
只有一個點的外分支度大於2時，則對於這個問題的一些特殊情形, 我們
提出了多項式時間的解法。最後, 對一般有根樹，我們設計了一個倍率為
mind f(d .. 1)(xd + 1)g 的近似演算法,其中, xd 是外分支度大於d 的節點
數。

The le dissemination problem under the trust model is rst studied by Ku et
al. (GLOBECOM 2012), who dened the trust relationship as a full binary
tree and design an algorithm to broadcast a le from the root to every user,
using an optimal number of rounds. Tien et al. (2016) extended the trust
relationship as a rooted binary DAG (Directed Acyclic Graph), and design
conceptually simpler algorithms to compute an optimal-round broadcast.
In this thesis, we further extend the trust model as a general-degree DAG.
We will show that when the maximum degree of DAG is at least 6, nding an
optimal schedule is a NP-hard problem. On the other hand, we also study a
special case when the trust relationship is a general tree with only one node
whose degree is greater than two. For such a problem, we show that we can
solve it in polynomial time. Finally, we design a mind f(d..1)(xd +1)g-ratio
approximation algorithm on general rooted tree T, where xd is the number
of nodes whose degree is greater than d in T.

1 Introduction 1
2 Related Work 3
3 Problem Denition 5
4 Preliminaries 7
5 Hardness of the ORFDS Problem on a Rooted DAG 9
5.1 The Problem is in NP . . . . . . . . . . . . . . . . . . . . . . 10
5.2 The Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.3 Necessity and Suciency . . . . . . . . . . . . . . . . . . . . . 14
5.3.1 Proof of the `only if' part . . . . . . . . . . . . . . . . 14
5.3.2 Proof of the `if' part . . . . . . . . . . . . . . . . . . . 16
6 The ORFDS Problem on a Rooted Tree 19
6.1 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.1.1 Only Root Has Degree Greater Than 2 . . . . . . . . . 19
6.1.2 Extension . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.2 Approximation Algorithm . . . . . . . . . . . . . . . . . . . . 23
7 Conclusion and Future Work 27

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