[Paper Review] Linear Network Code for Erasure Broadcast Channel with Feedback: Complexity and Algorithms

Chi Wan Sung, Linyu Huang+2 more|arXiv (Cornell University)|May 24, 2012

Cooperative Communication and Network Coding35 references4 citations

TL;DR

This paper proposes sparse linear network codes for erasure broadcast channels with feedback to minimize completion time and decoding delay. It proves that innovative encoding vectors are optimal for completion time, shows the existence problem is NP-complete when field size < number of users, and introduces an approximation algorithm for sparse code generation that reduces decoding complexity while maintaining low completion time, outperforming random linear network coding (RLNC) in decoding speed with feedback and higher encoding time.

ABSTRACT

This paper investigates the construction of linear network codes for broadcasting a set of data packets to a number of users. The links from the source to the users are modeled as independent erasure channels. Users are allowed to inform the source node whether a packet is received correctly via feedback channels. In order to minimize the number of packet transmissions until all users have received all packets successfully, it is necessary that a data packet, if successfully received by a user, can increase the dimension of the vector space spanned by the encoding vectors he or she has received by one. Such an encoding vector is called innovative. We prove that innovative linear network code is uniformly optimal in minimizing user download delay. When the finite field size is strictly smaller than the number of users, the problem of determining the existence of innovative vectors is proven to be NP-complete. When the field size is larger than or equal to the number of users, innovative vectors always exist and random linear network code (RLNC) is able to find an innovative vector with high probability. While RLNC is optimal in terms of completion time, it has high decoding complexity due to the need of solving a system of linear equations. To reduce decoding time, we propose the use of sparse linear network code, since the sparsity property of encoding vectors can be exploited when solving systems of linear equations. Generating a sparsest encoding vector with large finite field size, however, is shown to be NP-hard. An approximation algorithm that guarantee the Hamming weight of a generated encoding vector to be smaller than a certain factor of the optimal value is constructed. Our simulation results show that our proposed methods have excellent performance in completion time and outperforms RLNC in terms of decoding time.

Motivation & Objective

• To minimize user download delay in broadcast systems over erasure channels with feedback by ensuring each transmitted packet is innovative.
• To analyze the computational complexity of determining the existence of innovative encoding vectors, showing it is NP-complete when the finite field size is smaller than the number of users.
• To reduce decoding complexity of linear network coding by proposing sparse encoding vectors that exploit sparsity during Gaussian elimination.
• To design an approximation algorithm that guarantees the Hamming weight of generated encoding vectors is within a bounded factor of the optimal, for large finite fields.
• To compare the proposed scheme with RLNC and other broadcast codes, demonstrating tradeoffs in completion time, encoding, and decoding complexity.

Proposed method

• Uses innovative encoding vectors—those not in the span of previously received vectors—to minimize completion time, ensuring each transmission increases user reception dimension by one.
• Proves that when the finite field size q ≥ K (number of users), innovative vectors always exist, and RLNC can find them with high probability.
• Introduces an incremental null space basis computation method using Sherman-Morrison formula to efficiently verify innovativeness and update user state matrices.
• Proposes an approximation algorithm for generating the sparsest possible encoding vectors under large finite fields, with bounded performance relative to the optimal.
• For binary fields, a heuristic algorithm is designed to generate sparse encoding vectors efficiently.
• Employs feedback to allow the source to adaptively select encoding vectors that are innovative to the largest number of users, reducing redundant transmissions.

Experimental results

Research questions

• RQ1 Under what conditions do innovative encoding vectors exist in linear network codes for erasure broadcast channels?
• RQ2 What is the computational complexity of determining the existence of innovative encoding vectors when the finite field size is smaller than the number of users?
• RQ3 Can sparse linear network codes reduce decoding complexity without significantly increasing completion time compared to RLNC?
• RQ4 How can an approximation algorithm be designed to generate sparse encoding vectors with bounded Hamming weight relative to the optimal?
• RQ5 What tradeoffs exist between completion time, encoding time, and decoding time when using feedback and sparse coding schemes?

Key findings

• The existence of innovative encoding vectors is NP-complete when the finite field size is strictly smaller than the number of users, but always exists when q ≥ K.
• Random linear network coding (RLNC) achieves optimal completion time but incurs high decoding complexity due to solving dense linear systems.
• Sparse linear network codes significantly reduce decoding time by exploiting sparsity during Gaussian elimination, outperforming RLNC in decoding speed.
• The proposed approximation algorithm guarantees that the Hamming weight of generated encoding vectors is within a factor of the optimal, ensuring practical efficiency.
• For binary fields, the heuristic algorithm generates sparse encoding vectors that perform well in practice, reducing decoding time without increasing completion time.
• Simulation results show that the proposed scheme achieves lower decoding time than RLNC while maintaining competitive completion time, at the cost of higher encoding time and feedback requirement.