Time-optimal ranking algorithms on sorted matrices

Answering rank queries is a recurring operation in various application domains including geographic data processing, information retrieval, database design, information management, and medical image processing. Many of these applications involve data stored in a matrix satisfying a number of properties. One property that occurs time and again in applications specifies that the rows and the columns of the matrix are independently sorted. It is customary to refer to such a matrix as sorted. An instance of the Batched Ranking problem, (BR, for short) involves a sorted matrix A of items from a totally ordered universe, along with a collection Q of queries of the following type: for a query q_j one is interested in the number of items in A that are smaller than q_j. The BR problem asks for solving all queries in Q. In this work, we consider the BR problem in the following context: the matrix A is pretiled, one item per processor, onto an enhanced mesh of size √n×√n; the m queries are stored, one per processor, in the first m/√n columns of the platform. Our main contribution is twofold. First, we show that any algorithm that solves the BR problem must take at least Ω(log n+√m) time in the worst case. Second, we show that this time lower bound is tight on meshes of size √n×√n enhanced with multiple broadcasting, by exhibiting an algorithm solving the BR problem in O(log n+√m) time on such a platform