Encoding Two-Dimensional Range Top-k Queries

Abstract

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×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((k+1)nn)+2nm(m−1)+o(n)) bits, which generalizes the joint Cartesian tree of Golin et al. [TCS 2016]. Compared with trivial O(nmlgn)-bit encoding, our encoding takes less space when m=o(lgn). 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.