We consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering \({\text{Top-}}{k}\) queries. The aim is to obtain encodings that use space close to the information-theoretic lower bound, which can be constructed efficiently. For an \(m \times n\) array, with \(m \le n\), we first propose an encoding for answering 1-sided \({\textsf {Top}}{\text {-}}k{}\) queries, whose query range is restricted to \([1 \dots m][1 \dots a]\), for \(1 \le a \le n\). Next, we propose an encoding for answering for the general (4-sided) \({\textsf {Top}}{\text {-}}k{}\) queries that takes \((m\lg {{(k+1)n \atopwithdelims ()n}}+2nm(m-1)+o(n))\) 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 \({\textsf {Top}}{\text {-}}k{}\) queries, which show that our upper bound results are almost optimal.

我们考虑编码二维数组的问题，其元素来自全序，以回答\({\text{Top-}}{k}\)查询。目的是获得使用接近信息理论下界的空间的编码，可以有效地构造这些编码。对于\(m \times n\)数组，使用\(m \le n\)，我们首先提出一种编码来回答单边\({\textsf {Top}}{\text {-}}k{ }\)查询，其查询范围限制为\([1 \dots m][1 \dots a]\)，对于\(1 \le a \le n\)。接下来，我们提出了一种用于回答一般（4 边）\({\textsf {Top}}{\text {-}}k{}\)查询的编码，它需要\((m\lg {{(k+1)n \atopwithdelims ()n}}+2nm(m-1)+o(n))\)位，它概括了Golin 等人的联合笛卡尔树。[TCS 2016]。与普通的\(O(nm\lg {n})\)位编码相比，我们的编码在\(m = o(\lg {n})\)时占用的空间更少。除了编码的上限结果，我们还给出了回答 1 和 4 边\({\textsf {Top}}{\text {-}}k{}\)查询的编码下限，这表明我们的上限结果几乎是最优的。