Abstract
Author(s): Meng, C; Das, AK; Ramakrishnan, A; Jafar, SA; Markopoulou, A; Vishwanath, S | Abstract: We consider the problem of network coding across three unicast sessions over a directed acyclic graph, where the sender and receiver of each unicast session are both connected to the network via a single edge of unit capacity. We consider a network model in which the middle of the network can only perform random linear network coding, and restrict our approaches to precoding-based linear schemes, where the senders use precoding matrices to encode source symbols. We adapt a precoding-based interference alignment technique, originally developed for the wireless interference channel, to construct a precoding-based linear scheme, which we refer to as precoding-based network alignment scheme (PBNA). A primary difference between this setting and the wireless interference channel is that the network topology can introduce dependencies among the elements of the transfer matrix, which we refer to as coupling relations, and can potentially affect the achievable rate of PBNA. We identify all these coupling relations and interpret them in terms of network topology. We then present polynomial-time algorithms to check the presence of these coupling relations in a particular network. Finally, we show that, depending on the coupling relations present in the network, the optimal symmetric rate achieved by precoding-based linear scheme can take only three possible values, all of which can be achieved by PBNA.
Highlights
Ever since the development of network coding and its success in characterizing the achievable throughput for single multicast scenario [1] [2], there has been hope that the framework can be extended to characterize network capacity in other scenarios, namely inter-session network coding
There have been some successes in this domain, such as the derivation of a sufficient condition for linear network coding to achieve the maximal throughput in networks with multiple unicast sessions [3] [4]
Rate Optimality: We show that for the SISO scenarios where all senders are connected to all receivers via directed paths, depending on the coupling relations present in the network, there are only three possible optimal symmetric rates achieved by any precoding-based linear scheme, all of which are achievable through precodingbased network alignment scheme (PBNA)
Summary
Ever since the development of network coding and its success in characterizing the achievable throughput for single multicast scenario [1] [2], there has been hope that the framework can be extended to characterize network capacity in other scenarios, namely inter-session network coding. An important difference between the SISO scenario and the wireless interference channel is that there may be algebraic dependencies, which we refer to as coupling relations, between elements of the transfer matrix, which we refer to as transfer functions These are introduced by the network topology and may affect the achievable rate of PBNA [12]. Rate Optimality: We show that for the SISO scenarios where all senders are connected to all receivers via directed paths, depending on the coupling relations present in the network, there are only three possible optimal symmetric rates achieved by any precoding-based linear scheme (namely 1/3, 2/5 and 1/2), all of which are achievable through PBNA. In Appendix E, we present a comparison between routing and PBNA
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