An Efficient Algorithm for Optimally Solving a Shortest Vector Problem in Compute-and-Forward Protocol Design

TL;DR

This paper introduces a highly efficient algorithm for solving the shortest vector problem in compute-and-forward protocols, significantly reducing computational complexity while ensuring optimal coefficient vector selection.

Contribution

The paper presents a novel, low-complexity algorithm that transforms the quadratic form into an SVP and efficiently finds the optimal coefficient vector using modified sphere decoding techniques.

Findings

Algorithm achieves $igO(n^{1.5}P^{0.5})$ complexity for Gaussian channels.

Outperforms existing optimal algorithms in efficiency and speed.

Can be adapted for list output of multiple candidate vectors.

Abstract

We consider the problem of finding the optimal coefficient vector that maximizes the computation rate at a relay in the compute-and-forward scheme. Based on the idea of sphere decoding, we propose a highly efficient algorithm that finds the optimal coefficient vector. First, we derive a novel algorithm to transform the original quadratic form optimization problem into a shortest vector problem (SVP) using the Cholesky factorization. Instead of computing the Cholesky factor explicitly, the proposed algorithm realizes the Cholesky factorization with only $\bigO(n)$ flops by taking advantage of the structure of the Gram matrix in the quadratic form. Then, we propose some conditions that can be checked with $\bigO(n)$ flops, under which a unit vector is the optimal coefficient vector. Finally, by taking into account some useful properties of the optimal coefficient vector, we modify the Schnorr-Euchner search algorithm to solve the SVP. We show that the estimated average complexity of our new algorithm is $\bigO(n^{1.5}P^{0.5})$ flops for i.i.d. Gaussian channel entries with SNR $P$ based on the Gaussian heuristic. Simulations show that our algorithm is not only much more efficient than the existing ones that give the optimal solution, but also faster than some best known suboptimal methods. Besides, we show that our algorithm can be readily adapted to output a list of $L$ best candidate vectors for use in the compute-and-forward design. The estimated average complexity of the resultant list-output algorithm is $\bigO\left(n^{1.5}P^{0.5}\log L + nL\right)$ flops for i.i.d. Gaussian channel entries.