Local-search (LS)-based k-means clustering is a widely used method for solving the k-means problem and has attracted considerable academic attention since its inception. Previous research focused on optimizing LS steps to theoretically minimize clustering loss, leading to the development of various LS-type algorithms. This study seeks to improve the efficiency of state-of-the-art LS-type methods in high-dimensional spaces without altering the clustering outcomes. We identify that LS-type methods inherently involve redundant distance computations, which suggests potential efficiency gains through the elimination of these redundancies. To achieve this, we propose a method that utilizes lower-bound (LB) strategies to avoid unnecessary distance calculations, and introduce two effective LBs based on partial distance and segment mean. Our approach, combined with state-of-the-art LS-type algorithms, LS++ and LSDS++, is empirically validated. Experimental results show that integrating the proposed LB strategies reduces high-dimensional distance computations and results in faster performance compared to the state-of-the-art. Code is available at GitHub: https://github.com/pcjdhhhh/Local-Search-based-k-means/ .