點對點（P2P）技術可用於減輕資料傳輸時伺服器負擔以及減少平均下載時間，目前已有許多有效率且安全的方案。點對點的網絡普遍假設每個節點都可以傳送檔案給其他任何節點或從其他任何節點接收檔案，然而在現實生活中，檔案只能根據現在參與的節點之間的信任關係來傳收。

Ku等作者（GLOBECOM 2012）最先研究在有階層式信任關係下的最佳檔案傳輸問題。他們發現了在任何n個節點的完全二元樹狀（即每個內部節點都有兩個子節點）的信任關係下皆存在一個只需要 log2 n 回合的最佳檔案傳輸排程，且提出了一個線性時間的演算法去計算此最佳排程。

在本論文中，我們延伸了這個檔案傳輸問題，並把信任關係推廣至廣義的二元樹形式。也就是說，每個內部節點並沒有被限制一定要有兩個子節點。我們證明了在廣義的二元樹下，亦存在只需要 log2 n 個回合的的最佳檔案傳輸排程，而且也提出了可在線性時間內計算出此最佳排程的演算法。雖然我們延伸了Ku 等作者的結果，但是兩者之證明方法以及演算法之中心思考方式均截然不同。此外，我們也考慮了在平行計算模型下如何找到最佳排程，並提出了O(h log2 n) 的演算法，其中h是二元樹的高度。最後，我們發現了用以上提出的演算法皆可解決在有根有向無環圖的信任關係下的最佳排程問題。

Peer-to-Peer (P2P) technology has emerged as a solution of the file dissemination to lessen server load and reduce average download time. Many efficient and secure solutions were proposed. Yet, in a peer-to-peer network,it is generally assumed that every node can send and receive files to and from any other node, while in practice, files can only be transmitted according to certain trust relationship among participating peers.

Ku~et~al.~(GLOBECOM 2012) first studied the file dissemination problem under hierarchical trust relationship.
They showed that when the trust relationship is defined as a rooted full binary tree, then there exists an optimal schedule for file dissemination taking $\lceil \log_2 n \rceil$ rounds, where $n$ is the total number of nodes including the source and destinations of broadcasting.
Furthermore, they devised a linear-time algorithm to compute such a schedule.

In this thesis, we extend the file dissemination problem with the trust relationship in the form of general binary tree, i.e., each internal node is not restricted to have exactly two children.
We show that an optimal schedule for file dissemination remains $\lceil \log_2 n \rceil$ rounds, irrespective of the tree topology, and such a schedule can be computed in linear time.
While we are extending Ku~et~al.'s results, our algorithm is based on a completely different approach.
We have also considered the case of finding such an optimal schedule in the parallel setting, and propose an algorithm with parallel time complexity $O(h \log^{2} n)$, where $h$ denotes height of the tree.
Finally, we show that the sequential algorithm and the parallel algorithm work for the case when the trust relationship is a rooted DAG such that the out-degree of each node is bounded by two.

[1] M. Castro, P. Druschel, A. Kermarrec, A. Nandi, A. Rowstron, and A. Singh, \Splitstream: High-Bandwidth Multicast in Cooperative Environments," in Proceedings of ACM
SOSP, pp. 298{313, 2003.
[2] B. Cohen, \Incentives Build Robustness in Bittorrents," in Workshop on Economics of Peer-to-Peer Systems, 2003.
[3] T. T. Do, K. A. Hua, and M. A. Tantaoui, \Robust Video-On-Demand Streaming in Peer-to-Peer Environments," Computer Communications, 31(3):506{519, 2008.
[4] Y. Duan and J. Canny, \Scalable Secure Bidirectional
Group Communication," in Proceedings of IEEE INFOCOM, pp. 875{883, 2007.
[5] A. M. Farley, \Broadcast Time in Communication Net-
works," SIAM Journal on Applied Mathematics, 39(2):385{
390, 1980.
[6] P. Ganesan and M. Seshadri, \On Cooperative Content Dis-
tribution and the Price of Barter," in Proceedings of ICDCS,
pp. 81{90, 2005.
[7]C. Gkantsidis, T. Karagiannis, P. Rodriguez, and M. Vo-
jnovi, \Planet Scale Software Updates," Proceedings of ACM
SIGCOMM, pp. 423{434, 2006.
[8] Y. Guo, K. Suh, J. Kurose, and D. Towsley, \P2Cast: Peer-to-Peer Patching for Video On Demand Service," Multimedia Tools and Applications, 33(2):109{129, 2007.
[9] J. Hromkovic, R. Klasing, B. Monien, and R. Peine, \Dis-
semination of Information in Interconnection Networks,"
Combinatorial Network Theory, 125{212, 1996.
[10] B. Knutsson, H. Lu, W. Xu, and B. Hopkins, \Peer-to-Peer
Support for Massively Multiplayer Games," in Proceedings
of IEEE INFOCOM, pp. 107, 2004.
[11] D. Kostic, A. C. Snoeren, A. Vahdat, R. Braud, C. Killian,J. W. Anderson, J. Albrecht, A. Rodriguez, and E. Vandekieft, \High-Bandwidth Data Dissemination for Large-
Scale Distributed Systems," ACM Transactions on Com-
puter Systems (TOCS), 26(1):Article 3, 2008
[12] C.-F. Ku, K.-H. Yang, and J.-M. Ho, \An Optimal Schedul-
ing for File Dissemination under a Full Binary Tree of
Trust Relationship," in Proceedings of IEEE GLOBECOM,
pp. 751{757, 2012.
[13] J. Mundinger, R. R. Weber, and G. Weiss \Optimal
scheduling of peer-to-peer le dissemination," Journal of
Scheduling, 11(2[14] X. Yang and G. de Veciana, \Service Capacity of Peer-
to-Peer Networks," in Proceedings of IEEE INFOCOM,
pp. 2242{2252, 2004.
[15] Z. Zhou and D. Huang \An Optimal Key Distribution
Scheme for Secure Multicast Group Communication," in
Proceedings of IEEE INFOCOM, pp. 1-5, 2010.