Near-Optimal and Efficient Encoding for Two-Dimensional Range Minimum Queries

2026-07-05

AI Summary

This study addresses the problem of efficiently answering two-dimensional range minimum queries (2D RMQs) in the encoding model, where the original array is not accessible. The authors propose a novel encoding scheme based on hierarchical encoding, recursive partitioning, and parameterized indexing, complemented by an information-theoretic lower-bound analysis to guide data structure design. Their approach yields the first 2D RMQ encoding structure that simultaneously achieves near-optimal space complexity of $O(\kappa mn(\log m + \log\log n))$ bits and sub-logarithmic query time of $O(\log^{1/\kappa} n)$, where $\kappa \in [1, \log\log n]$ provides a flexible trade-off between space and time. This result overcomes longstanding limitations of prior methods concerning either query efficiency or parallelizability.

Abstract

We consider the 2D RMQ encoding problem: given an $m\times n$ array of $mn$ elements over a total order, encode it such that, for any query rectangle, the position of its maximum element can be reported without accessing the original array. For $m \le n$, it is known how to encode the array in $O(mn \min\{m, \log n\})$ bits with $O(1)$-time queries [Brodal et al., Algorithmica 2012], and also how to obtain an asymptotically optimal encoding consisting of $O(mn \log m)$ bits [Brodal et al., ESA 2013]. However, the latter approach does not prove any guarantee on the query time, and it appears to be inherently sequential: it requires scanning the whole encoding to answer a query. We design a different encoding that uses near-optimal space while allowing for efficient queries. More concretely, for every parameter $κ\in[1, \log\log n]$, our encoding uses $O(κmn(\log m+\log\log n))$ bits and answers 2D RMQ queries in $O(\log^{1/κ}n)$ time.

Problem

• 2D RMQ
• encoding
• range minimum query
• space-efficient
• query efficiency

Innovation

• 2D RMQ
• near-optimal encoding
• efficient query
• space-time tradeoff
• range minimum query