Upper and lower bounds for selection on the mesh

A distance-optimal algorithm for selection on the mesh has proved to be elusive, although distance-optimal algorithms for the related problems of routing and sorting have recently been discovered. In this paper, we explain, using the notion of adaptiveness, why techniques used in the currently best selection algorithms cannot lead to a distance-optimal algorithm. We also present the first algorithm for selection that has distance-optimal performance on average. For worst-case inputs, we apply new techniques to improve the previous best upper bound of 1.22n of Kaklanaanis et al. to 1.15n. This improvement is obtained in part by increasing the adaptiveness of previous algorithms