Encoding two-dimensional range top-k queries

TL;DR

This paper develops space-efficient encodings for answering two-dimensional range top-k queries, providing near-optimal solutions for both 1-sided and 4-sided queries with theoretical bounds.

Contribution

It introduces new encoding schemes for 2D range top-k queries that are close to the information-theoretic lower bounds, improving over trivial methods.

Findings

Proposed an encoding for 1-sided top-k queries with space efficiency.

Developed a general encoding for 4-sided top-k queries with near-optimal space.

Established lower bounds showing the near-optimality of the proposed encodings.

Abstract

We consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering \topk{} 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 \topk{} 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) \topk{} queries that takes $(m\lg{{(k+1)n \choose n}}+2nm(m-1)+o(n))$ bits, which generalizes the \textit{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 \topk{} queries, which show that our upper bound results are almost optimal.