The Range 1 Query (R1Q) Problem

We define the range 1 query (R1Q) problem as follows. Given a d-dimensional (d ≥ 1) input bit matrix A, preprocess A so that for any given region \(\mathcal{R}\) of A, one can efficiently answer queries asking if \(\mathcal{R}\) contains a 1 or not. We consider both orthogonal and non-orthogonal shapes for \(\mathcal{R}\) including rectangles, axis-parallel right-triangles, certain types of polygons, and spheres. We provide space-efficient deterministic and randomized algorithms with constant query times (in constant dimensions) for solving the problem in the word RAM model. The space usage in bits is sublinear, linear, or near linear in the size of A, depending on the algorithm.

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