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

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 $\kappa\in[1, \log\log n]$, our encoding uses $O(\kappa mn(\log m+\log\log n))$ bits and answers 2D RMQ queries in $O(\log^{1/\kappa}n)$ time.