Bounding multiple unicasts through index coding and Locally Repairable Codes
Bounding multiple unicasts through index coding and Locally Repairable Codes
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通过索引编码和本地可修复代码限制多个单播
DOI：
10.1109/isit.2014.6874842
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
A. Dimakis
中科院分区：
文献类型：
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作者：
Karthikeyan Shanmugam;A. Dimakis
建立了线性索引编码与局部可修码之间的对偶关系。具体来说，我们表明，一个自然的扩展LRC，我们称之为广义局部可修码（GLCR）是完全对偶的线性索引码。在GLRC中，每个节点都可以从其他节点的特定集合中解码，并且这些集合导出可恢复性有向图。我们表明，对偶线性子空间的GLRC是一个解决方案的索引编码的情况下，边信息图是这个GLRC可恢复图。我们表明，GLRC率是相当于互补索引编码率，即编码节省的传输次数。我们的第二个结果使用这种对偶建立一个新的上界的多单播网络编码问题。在多单播网络编码中，我们给定一个有向无环图和r个源，它们希望向r个相应的目的地发送独立的消息。我们的新上限是有效的计算，并依赖于一个强大的近似结果互补索引编码。我们相信，我们的界限可能会导致多个单播网络编码的近似保证，如果一个合理的连接，我们的状态是验证。
We establish a duality result between linear index coding and Locally Repairable Codes (LRCs). Specifically, we show that a natural extension of LRCs we call Generalized Locally Repairable Codes (GLCRs) are exactly dual to linear index codes. In a GLRC, every node is decodable from a specific set of other nodes and these sets induce a recoverability directed graph. We show that the dual linear subspace of a GLRC is a solution to an index coding instance where the side information graph is this GLRC recoverability graph. We show that the GLRC rate is equivalent to the complementary index coding rate, i.e. the number of transmissions saved by coding. Our second result uses this duality to establish a new upper bound for the multiple unicast network coding problem. In multiple unicast network coding, we are given a directed acyclic graph and r sources that want to send independent messages to r corresponding destinations. Our new upper bound is efficiently computable and relies on a strong approximation result for complementary index coding. We believe that our bound could lead to an approximation guarantee for multiple unicast network coding if a plausible connection we state is verified.