The Approximating and Eliminating Search Algorithm (AESA) can currently be considered as one of the most efficient procedures for finding Nearest Neighbours in Metric Spaces where distances computation is expensive. One of the major bottlenecks of the AESA, however, is its quadratic preprocessing time and memory space requirements which, in practice, can severely limit the applicability of the algorithm for large sets of data. In this paper a new version of the AESA is introduced which only requires linear preprocessing time and memory. The performance of the new version, referred to as ‘Linear AESA’ (LAESA), is studied through a number of simulation experiments in abstract metric spaces. The results show that LAESA achieves a search performance similar to that of the AESA, while definitely overcoming the quadratic costs bottleneck.