A graph minor perspective to network coding: Connecting algebraic coding with network topologies

Abstract

Abstract-Network Coding encourages information coding across a communication network. While the necessity, benefit and complexity of network coding are sensitive to the underlying graph structure of a network, existing theory on network coding often treats the network topology as a black box, focusing on algebraic or information theoretic aspects of the problem. This work aims at an in-depth examination of the relation between algebraic coding and network topologies. We mathematically establish a series of results along the direction of: if network coding is necessary/beneficial, or if a particular finite field is required for coding, then the network must have a corresponding hidden structure embedded in its underlying topology, and such embedding is computationally efficient to verify. Specifically, we first formulate a meta-conjecture, the NC-Minor Conjecture, that articulates such a connection between graph theory and network coding, in the language of graph minors. We next prove that the NC-Minor Conjecture is almost equivalent to the Hadwiger Conjecture, which connects graph minors with graph coloring. Such equivalence implies the existence of K4, K5, K6, and K O(q/ log q) minors, for networks requiring F3, F4, F5 and Fq, respectively. We finally prove that network coding can make a difference from routing only if the network contains a K4 minor, and this minor containment result is tight. Practical implications of the above results are discussed.

Information multicast in (pseudo-)planar networks: Efficient network coding over small finite fields

“…8 Remove the source S and let G1 denote the graph obtained here. 9 Color the graph G1 with 3 colors [19].…”

“…For the subtree graph of a multicast network G, contract each subtree to a node and connect the two nodes if the two corresponding subtrees share a common leaf in G. We refer to the resulting graph as the subtree-node graph of G. A recent work of Yin et al [9] show that if a multicast network G requires GF (3), then the subtree-node graph of G must contain a K 4 minor. Below we first prove that the subtree-node graph of a terminal-coface network can not be K 4 .…”

The Adaptive Network Coding-Routing (ANCR) Based on Graph Minors for Multicast Networks

“…1. As soon as the decomposition process is finished, the source can build up a graph minor, which contains an appropriate group of nodes used for XORing, if a graph minor is K 4 as claimed in [7]. The result shows packets generated only from nodes in the same minor that can be allowed to apply network coding before relaying or not shown in Fig.…”

“…In this work, we consider a butterfly network for 2-minimal multicast communications and propose the algorithm called "adaptive network coding-routing (ANCR)," where all relays are capable to either forward data to the best route using AODV and AODV-BR routing protocol or algebraic compute packets before routing. Our scheme is inspired by [7] introduced and proved that a K 4 graph minor of any graph topology can reduce computation time of network coding. They introduced the relation between algebraic coding and network topologies and claimed that network coding can be applied and make a difference from routing only if the network topology contains a K 4 minor.…”