Encoding Two-Dimensional Range Top-k Queries

We consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering Top-k queries. The aim is to obtain encodings that use space close to the information-theoretic lower bound, which can be constructed efficiently. For an m x n array, with m <= n, we first propose an encoding for answering 1-sided Top-k queries, whose query range is restricted to [1. . . m][1. . . a], for 1 <= a <= n. Next, we propose an encoding for answering for the general (4-sided) Top-k queries that takes (mlg ( [GRAPHICS] ) + 2nm(m - 1) + o(n)) bits, which generalizes the joint Cartesian tree of Golin et al. [TCS 2016]. Compared with trivial O(nm lg n) -bit encoding, our encoding takes less space when m = o(lg n). In addition to the upper bound results for the encodings, we also give lower bounds on encodings for answering 1 and 4-sided Top-k queries, which show that our upper bound results are almost optimal.