Network Coding in Minimal Multicast Networks

Abstract

We investigate the network coding problem in a minimal size of a finite field over which a linear network code certain class of minimal multicast networks. In a multicast coding exists for a certain multicast problem is NP-hard. network, a source S needs to deliver h symbols, or packets, to aIn this work, we focus on the network coding problem set of destinations T over an underlying communication network for minimal multicast networks. A coding network (G, S, T) modeled by a graph G. A coding network is said to be h-minimal i if it can deliver h symbols from S to the destination nodes, while is said to be h-minimal if it can deliver h packets from any proper subnetwork of G can deliver at most h -1 symbols S to T, while any proper subnetwork of G can deliver to the set of destination nodes. This problem is motivated by at most h -1 packets. The problem is motivated by the the requirement to minimize the amount of network resources need of network service providers to minimize the amount allocated for a multicast connections.. Wllocaed how ta multicast networksohav of network resources allocated for individual multicast We show that, surprisingly, minimal multicast networks have concin. Ined miia. utcs ntok nld unique properties that distinguish them from the general case of connections. Indeed, minimal multcast networks include multicast networks. In particular, we show that it is possible to only links that are essential for delivery of h packets to determine whether a 2-minimal network has a routing solution all T terminals, which minimizes the cost of establishing a (i.e., a solution without encoding nodes) in polynomial time, while multicast connection. this problem is NP-hard in general. In addition, we show that if a 2-minimal network is planar, then the minimum size of the required field for linear network codes is at most 3. Also, weContributions. The contribution of our paper can be suminvestigate several structural properties of 2-minimal networks marized as follows. First, we analyze the complexity of and generalize our results for h > 2. deciding whether a given multicast problem admits a pure routing solution (i.e., a solution that does not require networkcoding). We show that this problem can be solved in linear time for 2-minimal coding networks. For the general case of A fundamental problem in the design of communication non-minimal coding networks, this problem was shown to be networks is to deliver information between the source and the NP-hard [12]. We present here another proof of this result destination nodes. Recently, it was shown that the information based on a reduction from the problem of vertex coloring of delivery can be facilitated by employing the novel technique multigraphs. of network coding [5]. The main idea of network codingWe also show that all network coding problems in 2is to allow intermediate nodes in the network to generate minimal networks have a similar structure. Specifically, any new packets by mixing the information received over their such problem can be reduced to ...

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

“…For h = 2, we assume G is a minimal network supporting h = 2 [15], i.e., any edge removal makes a multicast rate 2 infeasible. For such edge-minimal networks, we sometimes consider its orientation in which the max-flow from the source to each receiver is 2, and refer to the in-degree and out-degree of nodes in such an orientation.…”

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

“…Combined with Kuratowski's Theorem [6], this further implies that coding over F 3 suffices for all planar networks. While there exist proofs for the latter result that exploit planarity of the network [5], [8], our result reveals that whether planarity holds is actually not important, and planar networks enjoy the sufficiency of F 3 because they form a special class of K 5 -free networks. Our result also reveals that the de facto standard of using F 2 8 and F 2 16 in network coding implementations is an overkill, in the sense that no conceivable real-world network can have a so large clique minor.…”

“…A multicast network is h-minimal if it can deliver h flows to all the receivers but not with any of its links removed. The in-degree of a node is at most h in such a network [8].…”

“…To study this conjecture, we focus on the basic scenario of multicast, where the source has two information flows to disseminate. A multicast network G is 2-minimal if a multicast rate 2 is feasible in G but infeasible with any edge in G removed [8]. 2-minimal networks are easy to analyze, yet fundamental: they lead to the largest known throughput gap between network coding and routing [9], and require a fullfledged suite of coding operations based on unbounded field sizes, for unlimited number of receivers [4], [9].…”

Min-cost multicast networks in Euclidean space

“…In the literature of network coding, the case of two integral flows has attracted considerable research interests [7] [8]. It represents the most basic scenario where network coding can make a difference from routing.…”