完备k-树中的线广播
Line-Broadcasting in Complete k-Trees
摘要：
连接图$G=(V,E)$中的线路广播模型，即$|V|=n$，是这样一种模型，其中一个称为广播的{ \it始发者}的顶点持有必须通过在图上放置一系列调用而被发送到图的所有顶点的消息。在这个模型中，一个消息顶点可以在一个时间单位内通过任意长度的路径传输消息，只要两个传输不同时使用同一条边。Farley \cite{f}已经证明，从树中的任何发起者（因此在任何连接的无向图中）开始，该过程最多在$\lceil \log_{2}n \rceil$时间单位内完成。并且从任何顶点广播一个消息的成本至多为$(n-1) \lceil \log_{2}n \rceil$。在本文中，我们提出了成本的下限和上限广播一个消息在一个完整的$k-$树，从任何顶点使用线广播模型。我们证明，如果$B(u)$是最小成本的图$G=(V,E)$从一个顶点$u \in V$使用线广播模型广播，那么$(1+o(1))n \le B(u) \le (2+o(1))n$，其中$u$是任何顶点在一个完整的$k$树。此外，对于某些条件，$B(u) \le (2-o(1))n$。
英文摘要：
A line-broadcasting model in a connected graph $G=(V,E)$, $|V|=n$, is a model in which one vertex, called the {\it originator} of the broadcast holds a message that has to be transmitted to all vertices of the graph through placement of a series of calls over the graph. In this model, an informed vertex can transmit a message through a path of any length in a single time unit, as long as two transmissions do not use the same edge at the same time. Farley \cite{f} has shown that the process is completed within at most $\lceil \log_{2}n \rceil$ time units from any originator in a tree (and thus in any connected undirected graph). and that the cost of broadcasting one message from any vertex is at most $(n-1) \lceil \log_{2}n \rceil$. In this paper, we present lower and upper bounds for the cost to broadcast one message in a complete $k-$tree, from any vertex using the line-broadcasting model. We prove that if $B(u)$ is the minimum cost to broadcast in a graph $G=(V,E)$ from a vertex $u \in V$ using the line-broadcasting model, then $(1+o(1))n \le B(u) \le (2+o(1))n$, where $u$ is any vertex in a complete $k$-tree. Furthermore, for certain conditions, $B(u) \le (2-o(1))n$.
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