arXiv 1202.07472012-02cs.ITmath.IT

网络编码复杂性的图论方法

A Graph Theoretical Approach to Network Encoding Complexity

Li Xu, Weiping Shang, Guangyue Han

备注：: 44 pages, 22 figures

摘要：

考虑具有源$S_1, S_2,..., S_l$和不同宿$R_1, R_2,..., R_l$的非循环有向网络$G$。对于$i=1, 2,..., l$，令$c_i$表示$S_i$和$R_i$之间的最小切割。然后，根据Menger定理，存在一组从$S_i$到$R_i$的边不交的$c_i$路，本文称之为从$S_i$到$R_i$的Menger路组.虽然在同一组中，它们是边缘不相交的，但来自不同组的门格尔路径可能必须彼此合并。通过适当地选择Menger路，不同Menger路组之间的合并数总是有界的，且与$G$的规模和拓扑无关.当所有$S_i$都不同时，所有上述网络的最紧密的这种常数用$\mathcal{M}(c_1, c_2,..., c_2)$表示，而当所有$S_i$实际上都相同时，用$\mathcal{M}^*(c_1, c_2,..., c_2)$表示。结果表明，$\mathcal{M}$和$\mathcal{M}^*$与多播网络、双向网络和多会话单播网络的网络编码复杂度密切相关。使用这种连接，我们计算在本文中的一些确切的值和边界的网络编码复杂性使用图论的方法。

英文摘要：

Consider an acyclic directed network $G$ with sources $S_1, S_2,..., S_l$ and distinct sinks $R_1, R_2,..., R_l$. For $i=1, 2,..., l$, let $c_i$ denote the min-cut between $S_i$ and $R_i$. Then, by Menger's theorem, there exists a group of $c_i$ edge-disjoint paths from $S_i$ to $R_i$, which will be referred to as a group of Menger's paths from $S_i$ to $R_i$ in this paper. Although within the same group they are edge-disjoint, the Menger's paths from different groups may have to merge with each other. It is known that by choosing Menger's paths appropriately, the number of mergings among different groups of Menger's paths is always bounded by a constant, which is independent of the size and the topology of $G$. The tightest such constant for the all the above-mentioned networks is denoted by $\mathcal{M}(c_1, c_2,..., c_2)$ when all $S_i$'s are distinct, and by $\mathcal{M}^*(c_1, c_2,..., c_2)$ when all $S_i$'s are in fact identical. It turns out that $\mathcal{M}$ and $\mathcal{M}^*$ are closely related to the network encoding complexity for a variety of networks, such as multicast networks, two-way networks and networks with multiple sessions of unicast. Using this connection, we compute in this paper some exact values and bounds in network encoding complexity using a graph theoretical approach.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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