Embodiments of the present invention provide efficient greedy LLL algorithms that not only converge faster but also exhibit much lower complexity than the existing greedy LLL variants while similar error performance is maintained. First, a relaxed Lovász condition is designed for searching the candidate set of LLL iterations with column swap operations. This relaxation does not need size reduction operations so that it can save complexity compared to the existing greedy LLL algorithms. Further, a relaxed criterion of the decrease in LLL potential is designed to select the optimal one in the candidate set of LLL iterations, which also exhibits lower complexity than the existing greedy LLL algorithms. Furthermore, simulations show that the inventive algorithm needs less LLL iterations compared to the existing greedy LLL algorithms.