在通訊網路模型中，點對點 (P2P) 技術為傳統集中式網路的替代方案，可降低
頻寬、縮短傳輸時間、增加網路規模的彈性、及提供更多自我錯誤修補的機制。
然而點與點之間的誠實信任成為了新的隱憂，Ku 等作者 (2012) 最先研究在階
級式信任網路中的檔案傳輸問題，提出了在信任階級模型為有根滿二元樹時，
線性時間的最佳排程演算法。隨後，Tien 等作者 (2019) 將信任階級模型延伸
為有根的二元有向無環圖，並提出了線性時間的最佳排程演算法。另一方面，
Lin(2017) 提出當信任階級模型為最大外分支度 6 的有向無環圖時，最佳排程問
題將為 NP 困難問題。
本論文將信任階級模型進一步延伸為有根的二元一般圖，並提出可於線性時
間內算出最佳排程的演算法。接著我們引入了有更強傳輸能力的超級節點，並
且發現：在二元有向無環圖上，儘管只有兩種不同傳輸能力的節點，其最佳排
程問題已為 NP 困難問題；但當信任階級模型僅為路徑 (一元樹) 時，則可在多
項式時間內求得最佳排程。
總的來說，階級式信任關係的結構、節點的最大外分支度、與多樣的傳播能
力皆為影響檔案傳輸問題難度的重要參數。試以不同的參數組合來規範檔案傳
輸問題，並將之界定為多項式時間內可解或 NP 困難題型，將成為未來一個有
趣的研究方向。

Peer-to-Peer (P2P) technology was proposed as an alternative solution to the
traditional centralized communication network model, which has the advantages
of bandwidth reduction, shorter dissemination time, better scalability, and faulttolerance. Yet, this brings new concerns, such as trust. Ku et al. (2012) first
proposed the file dissemination problem under hierarchical trust relationship, and
showed that when the trust hierarchy is a rooted full binary tree, an optimal file
dissemination schedule can be computed in linear time. Tien et al. (2019) later
extended the result so that the trust hierarchy can be any rooted directed acyclic
graph (DAG) whose vertices have out-degree at most 2; the time to compute an
optimal file dissemination schedule remains to be linear. In contrast, Lin (2017)
showed that, if the maximum out-degree of a node is allowed to be up to 6,
the problem of finding an optimal file dissemination schedule in a rooted DAG
suddenly becomes NP-hard.
In this thesis, we extend the file dissemination problem to consider trust hierarchy to be a general directed graph, with out-degree of every node at most 2,
and show how to find an optimal file dissemination in linear time. Next, we introduce the concept of superior nodes, which have stronger dissemination power
than others. We show that allowing superior nodes will make the file dissemination problem NP-Hard, even when there are only two types of dissemination
powers, and the trust hierarchy is a rooted DAG whose nodes have out-degree
at most 2. However, if we restrict the trust hierarchy to be a path, the problem
becomes polynomial-time solvable.
In summary, topology of the trust hierarchy, maximum degree of the nodes,
and diversity of dissemination powers are some key parameters that control the
hardness of the file dissemination problem. Finding the boundary classes (i.e., the
combinations of these parameters that separate between polynomial-time solvable
instances and NP-hard instances) will be an interesting direction to investigate.

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