Entropy-bounded representation of point grids

Abstract

We give the first fully compressed representation of a set of m points on an n×n grid, taking H+o(H) bits of space, where H=lg(n2m) is the entropy of the set. This representation supports range counting, range reporting, and point selection queries, with complexities that go from O(1) to O(lg2n/lglgn) per answer as the entropy of the grid decreases. Operating within entropy-bounded space, as well as relating time complexity with entropy, opens a new line of research on an otherwise well-studied area.