A Time-Optimal Multiple Search Algorithm on Enhanced Meshes,with Applications

Given a sorted sequence A = a₁, a₂, ..., a_n of items from a totally ordered universe, along with an arbitrary sequence Q = q₁, q₂, ..., q_m (1 ≤ m ≤ n) of queries, the multiple search problem involves computing for every q_j (1 ≤ j ≤ m) the unique a_i for which a_i−1 ≤ q_j < a_i. In this paper we propose a time-optimal algorithm to solve the multiple search problem on meshes with multiple broadcasting. More specifically, on a formula] × formula] mesh with multiple broadcasting, our algorithm runs in formula] time which is shown to be time-optimal. We also present some surprising applications of the multiple search algorithm to computer graphics, image processing, robotics, and computational geometry.