Network Coding: A Computational Perspective

Abstract

In this work, we study the computational perspective of network coding, focusing on two issues. First, we address the computational complexity of finding a network code for acyclic multicast networks. Second, we address the issue of reducing the amount of computation performed by the network nodes. In particular, we consider the problem of finding a network code with the minimum possible number of encoding nodes, i.e., nodes that generate new packets by combining the packets received over incoming links. We present a deterministic algorithm that finds a feasible network code for a multicast network over an underlying graph G(V, E) in time O(|E|kh+|V|k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> +h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> (k+h)), where k is the number of destinations and h is the number of packets. This improves the best known running time of O(|E|kh+|V|k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (k+h)) of Jaggi et al. (2005) in the typical case of large communication graphs. In addition, our algorithm guarantees that the number of encoding nodes in the obtained network code is bounded by O(h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). Next, we address the problem of finding a network code with the minimum number of encoding nodes in both integer and fractional coding networks. We prove that in the majority of settings this problem is NP-hard. However, we show that if h=O(1) and k=O(1) and the underlying communication graph is acyclic, then there exists an algorithm that solves this problem in polynomial time.