Network coding: a computational perspective

IEEE Transactions on Information Theory, 2005

The famous max-flow min-cut theorem states that a source node can send information through a network ( ) to a sink node at a rate determined by the min-cut separating and . Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to re-encode the information they receive. We demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed. We give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity. We extend these algorithms to integer capacities and to codes that are tolerant to edge failures.

2006 IEEE Information Theory Workshop, 2006

We investigate the network coding problem in a certain class of minimal multicast networks. In a multicast coding network, a source S needs to deliver h symbols, or packets, to a set of destinations T over an underlying communication network modeled by a graph G. A coding network is said to be h-minimal if it can deliver h symbols from S to the destination nodes, while any proper subnetwork of G can deliver at most h − 1 symbols to the set of destination nodes. This problem is motivated by the requirement to minimize the amount of network resources allocated for a multicast connections.

2012 9th International Conference on Fuzzy Systems and Knowledge Discovery, 2012

The encoding complexity of network coding for single multicast networks has been intensively studied from several aspects: e.g., the time complexity, the required number of encoding links, and the required field size for a linear code solution. However, these issues as well as the solvability are less understood for networks with multiple multicast sessions. Recently, Wang and Shroff showed that the solvability of networks with two unit-rate multicast sessions (2-URMS) can be decided in polynomial time . In this paper, we prove that for the 2-URMS networks: 1) the solvability can be determined with time O(|E|); 2) a solution can be constructed with time O(|E|); 3) an optimal solution can be obtained in polynomial time; 4) the number of encoding links required to achieve a solution is upper-bounded by max{3, 2N -2}; and 5) the field size required to achieve a linear solution is upper-bounded by max{2, ⌊ 2N -7/4 + 1/2⌋}, where |E| is the number of links and N is the number of sinks of the underlying network. Both bounds are shown to be tight.

2006

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.

IEEE Transactions on Information Theory, 2013

The encoding complexity of network coding for single multicast networks has been intensively studied from several aspects: e.g., the time complexity, the required number of encoding links, and the required field size for a linear code solution. However, these issues as well as the solvability are less understood for networks with multiple multicast sessions. Recently, Wang and Shroff showed that the solvability of networks with two unit-rate multicast sessions (2-URMS) can be decided in polynomial time [4]. In this paper, we prove that for the 2-URMS networks: 1) the solvability can be determined with time O(|E|); 2) a solution can be constructed with time O(|E|); 3) an optimal solution can be obtained in polynomial time; 4) the number of encoding links required to achieve a solution is upper-bounded by max{3, 2N − 2}; and 5) the field size required to achieve a linear solution is upper-bounded by max{2, ⌊ 2N − 7/4 + 1/2⌋}, where |E| is the number of links and N is the number of sinks of the underlying network. Both bounds are shown to be tight.

2006

Networking, IEEE/ACM Transactions on, 2003

We take a new look at the issue of network capacity. It is shown that network coding is an essential ingredient in achieving the capacity of a network. Building on recent work by Li et al., who examined the network capacity of multicast networks, we extend the network coding framework to arbitrary networks and robust networking. For networks which are restricted to using linear network codes, we find necessary and sufficient conditions for the feasibility of any given set of connections over a given network. We also consider the problem of network recovery for nonergodic link failures. For the multicast setup we prove that there exist coding strategies that provide maximally robust networks and that do not require adaptation of the network interior to the failure pattern in question. The results are derived for both delay-free networks and networks with delays.

2008

Network coding promises to significantly impact the way communications networks are designed, operated, and understood. This book presents a unified and intuitive overview of the theory, applications, challenges, and future directions of this emerging field, and is a must-have resource for those working in wireline or wireless networking. • Uses an engineering approach - explains the ideas and practical techniques • Covers mathematical underpinnings, practical algorithms, code selection, security, and network management • Discusses key topics of inter-session (non-multicast) network coding, lossy networks, lossless networks, and subgraph-selection algorithms Starting with basic concepts, models, and theory, then covering a core subset of results with full proofs, Ho and Lun provide an authoritative introduction to network coding that supplies both the background to support research and the practical considerations for designing coded networks. This is an essential resource for gradu...

IEEE Transactions on Information Theory, 2005

The famous max-flow min-cut theorem states that a source node can send information through a network ( ) to a sink node at a rate determined by the min-cut separating and . Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to re-encode the information they receive. We demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed. We give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity. We extend these algorithms to integer capacities and to codes that are tolerant to edge failures.

2007

Abstract Multicast is an important communication paradigm, also a problem well known for its difficulty (NP-completeness) to achieve certain optimization goals, such as minimum network delay. Recent advances in network coding has shed a new light onto this problem. In network coding, forwarding nodes can perform arbitrary operations on data received, other than forwarding or replicating, to enhance throughput of a multicast session.