We consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering Top-k\documentclass[12pt]{minimal}
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\begin{document}$${\text{Top-}}{k}$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$m \times n$$\end{document} array, with m≤n\documentclass[12pt]{minimal}
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\begin{document}$$m \le n$$\end{document}, we first propose an encoding for answering 1-sided Top-k\documentclass[12pt]{minimal}
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\begin{document}$${\textsf {Top}}{\text {-}}k{}$$\end{document} queries, whose query range is restricted to [1⋯m][1⋯a]\documentclass[12pt]{minimal}
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\begin{document}$$[1 \dots m][1 \dots a]$$\end{document}, for 1≤a≤n\documentclass[12pt]{minimal}
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\begin{document}$$1 \le a \le n$$\end{document}. Next, we propose an encoding for answering for the general (4-sided) Top-k\documentclass[12pt]{minimal}
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\begin{document}$${\textsf {Top}}{\text {-}}k{}$$\end{document} queries that takes (mlg(k+1)nn+2nm(m-1)+o(n))\documentclass[12pt]{minimal}
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\begin{document}$$(m\lg {{(k+1)n \atopwithdelims ()n}}+2nm(m-1)+o(n))$$\end{document} bits, which generalizes the joint Cartesian tree of Golin et al. [TCS 2016]. Compared with trivial O(nmlgn)\documentclass[12pt]{minimal}
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\begin{document}$$O(nm\lg {n})$$\end{document}-bit encoding, our encoding takes less space when m=o(lgn)\documentclass[12pt]{minimal}
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\begin{document}$$m = o(\lg {n})$$\end{document}. 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\documentclass[12pt]{minimal}
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\begin{document}$${\textsf {Top}}{\text {-}}k{}$$\end{document} queries, which show that our upper bound results are almost optimal.
