An efficient branch-and-bound algorithm for compute-and-forward

Abstract

Compute-and-forward is a framework for reliable physical layer network coding introduced by Nazer and Gastpar. Instead of decoding single messages, it decodes linear combinations of messages with the help of nested lattice codes. Nazer and Gastpar derived an achievable rate for each node depending on the channel coefficients and the desired equation coefficients. However, it is open how to choose the coefficient vector with the equation coefficients. We provide a branch-and-bound algorithm that calculates the coefficient vector, which results in the highest computation rate at a single node. We implemented the algorithm in Matlab and compared the number of iterations to the number of needed iterations if we use a complete search over all possible vectors.Index Terms-compute-and-forward, network coding, branch-and-bound algorithm

Compute-and-forward protocol design based on improved sphere decoding

“…• The branch-and-bound (BnB) algorithm [4]. • The method based on LLL lattice reduction algorithm [5], where δ is set as 0.75 since further increasing δ achieves little gain in rate but will greatly increase the running time.…”

“…The Fincke-Pohst method [2] was modified in [3] to solve a different but related problem, leading to the optimal coefficient vector and some other suboptimal vectors. A branch-and-bound algorithm, which uses part of the properties of the optimal vector, was used in [4]. But it appears that this algorithm is not efficient in this application.…”

Weak Secrecy in the Multiway Untrusted Relay Channel With Compute-and-Forward

“…Because all L users need to be able to decode all messages, we additionally require the rows of A to be linear independent of all vectors e where e is the unit vector with a one at the -th position and zeros elsewhere. This results in the following optimization problem There are several algorithms which find the best coefficient vector, e.g., [36]- [39]. These algorithms need to be extended to fulfill the following additional constraints:…”

Non-Cooperative Compute-and-Forward Strategies in Gaussian Multi-Source Multi-Relay Networks

“…This can be solved by several algorithms, e.g., [10]- [12]. Because we assume no cooperation between the relays, it is not guaranteed that the relay nodes decode linear independent linear combinations.…”

“…for the optimal equation in terms of achievable computation rate [10]- [12]. While most of these algorithms perform a local optimization, they ignore the network structure and the possible linear dependence of the equations at the final destination.…”