The encoding complexity of network coding

Abstract

Abstract-In the multicast network coding problem, a source needs to deliver packets to a set of terminals over an underlying communication network . The nodes of the multicast network can be broadly categorized into two groups. The first group incudes encoding nodes, i.e., nodes that generate new packets by combining data received from two or more incoming links. The second group includes forwarding nodes that can only duplicate and forward the incoming packets. Encoding nodes are, in general, more expensive due to the need to equip them with encoding capabilities. In addition, encoding nodes incur delay and increase the overall complexity of the network. Accordingly, in this paper, we study the design of multicast coding networks with a limited number of encoding nodes. We prove that in a directed acyclic coding network, the number of encoding nodes required to achieve the capacity of the network is bounded by 3 2 . Namely, we present (efficiently constructible) network codes that achieve capacity in which the total number of encoding nodes is independent of the size of the network and is bounded by 3 2 . We show that the number of encoding nodes may depend both on and by presenting acyclic coding networks that require ( 2 ) encoding nodes. In the general case of coding networks with cycles, we show that the number of encoding nodes is limited by the size of the minimum feedback link set, i.e., the minimum number of links that must be removed from the network in order to eliminate cycles. We prove that the number of encoding nodes is bounded by (2 + 1) 3 2 , where is the minimum size of a feedback link set. Finally, we observe that determining or even crudely approximating the minimum number of required encoding nodes is an -hard problem.

Communicating the sum of sources in a 3-sources/3-terminals network

“…We refer to such graphs as structured graphs. Our efficient reduction follows that appearing in [4], and has the additional following properties: (a)Ĝ is acyclic. (b) For every source (terminal) in G there is a corresponding source (terminal) inĜ.…”

“…(c) For any two edge disjoint paths P 1 and P 2 connecting a source terminal pair in G, there exists two vertex disjoint paths inĜ connecting the corresponding source terminal pair. (d) Any feasible network coding solution inĜ can be efficiently turned into a feasible network coding solution in G. Our reduction follows that appearing in [4] and is given here for completeness.…”

Min-cost multicast networks in Euclidean space

“…2 in an acyclic multicast network [9], we can show that the number of relay nodes is upper bounded in such networks in a similar way: in an h-minimal network, any relay node with in-degree larger than 1 must be a coding node and thus the total number of relays is bounded by h 3 (n − 1) 2 ; for the relay nodes with in-degree 1, they appear exactly once in each coding subtree. The number of coding forests, each of which is composed of coding subtrees of the same coding vector, is no more than n h , since a finite field of size n is sufficient and the number of different coding vectors is no more than n h .…”

“…While searching for the optimal solution, it is reasonable to consider only the h-minimal networks [9], where deleting any link will cause λ D (s, t) < h for some receiver t. According to Li, et al [5], for any h-minimal network, there is a linear network code where each link is assigned a global coding vector from the h dimension linear space F h . Therefore in a h-minimal network, each node has in-degree at most h, since the coding vector on an extra link would be linearly dependent with the other h coding vectors and thus redundant.…”

A path-oriented encoding evolutionary algorithm for network coding resource minimization

“…Only merging nodes are possible to become coding nodes. The number of coding links is used to estimate the amount of coding operations performed during the data transmission (Langberg et al, 2006). More descriptions can be found in Xing and Qu (2012 the data rate between s and t k in the NCM subgraph…”