The α-neighbor p-center problem (α-pCP) is an extension of the classical p-center problem. It aims to select p centers from a set of candidate centers to minimize the maximum distance between any client and its α service centers. In this paper, we propose a weighting-based fast local search algorithm called WFLS for solving α-pCP. First, WFLS converts the complex α-pCP into a series of decision subproblems by specifying the service radius, effectively mitigating the gradient vanishing issue during the search process, and introduces a new MIP model. Then, it addresses the simpliffed subproblems using a fast local search procedure with a swap-based neighborhood structure. WFLS adopts an efffcient weighting strategy, an incremental evaluation technique, a reffned-grained penaltybased neighborhood evaluation, and two scoring functions of neighborhood evaluation to accelerate and guide the search process. Computational experiments on 154 widely used public benchmark instances demonstrate that WFLS outperforms the state-of-the-art methods in the literature. Speciffcally, WFLS improves 69 previous best known results and matches the best know results for all the remaining ones in less time than other competitors.