A Deterministic Polynomial-Time Algorithm for Constructing a Multicast Coding Scheme for Linear Deterministic Relay Networks

Abstract

We propose a new way to construct a multicast coding scheme for linear deterministic relay networks. Our construction can be regarded as a generalization of the well-known multicast network coding scheme of Jaggi <etal xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> to linear deterministic relay networks and is based on the notion of flow for a unicast session that was introduced by the authors in earlier work. We present randomized and deterministic polynomial-time versions of our algorithm and show that for a network with <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$g$</tex></formula> destinations, our deterministic algorithm can achieve the capacity in <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\left \lceil \log (g+1)\right \rceil $</tex></formula> uses of the network and has the fastest construction time among algorithms for this problem.