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Entropy-Bounded Representation of Point Grids

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Arash Farzan, Travis Gagie, and Gonzalo Navarro

We give the first *fully compressed* representation of a set of *m*
points on an *n x n* grid, taking *H+o(H)* bits of space, where
*H = lg binomial(n^2,m)* 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(lg^2 n / lg lg n)* 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.