Nearly Tight Lower Bounds for Succinct Range Minimum Query

TL;DR

This paper establishes nearly tight lower bounds on the space complexity of data structures for the Range Minimum Query problem, matching the upper bounds up to polylogarithmic factors, thus nearly closing the gap in understanding optimal space-time trade-offs.

Contribution

It proves a nearly tight lower bound on space for RMQ data structures with given query time, improving previous bounds and matching upper bounds up to polylogarithmic factors.

Findings

Lower bound matches upper bounds up to polylogarithmic factors.

Space complexity must be at least $2n - 1.5\

n + n/(\

Abstract

Given an array of distinct integers $A[1\ldots n]$, the Range Minimum Query (RMQ) problem requires us to construct a data structure from $A$, supporting the RMQ query: given an interval $[a,b]\subseteq[1,n]$, return the index of the minimum element in subarray $A[a\ldots b]$, i.e. return $\text{argmin}_{i\in[a,b]}A[i]$. The fundamental problem has a long history. The textbook solution which uses $O(n)$ words of space and $O(1)$ time by Gabow, Bentley, Tarjan (STOC 1984) and Harel, Tarjan (SICOMP 1984) dates back to 1980s. The state-of-the-art solution is presented by Fischer, Heun (SICOMP 2011) and Navarro, Sadakane (TALG 2014). The solution uses $2n-1.5\log n+n/\left(\frac{\log n}{t}\right)^t+\tilde{O}(n^{3/4})$ bits of space and $O(t)$ query time, where the additive $\tilde{O}(n^{3/4})$ is a pre-computed lookup table used in the RAM model, assuming the word-size is $\Theta(\log n)$ bits. On the other hand, the only known lower bound is proved by Liu and Yu (STOC 2020). They show that any data structure which solves RMQ in $t$ query time must use $2n-1.5\log n+n/(\log n)^{O(t^2\log^2t)}$ bits of space, assuming the word-size is $\Theta(\log n)$ bits. In this paper, we prove nearly tight lower bound for this problem. We show that, for any data structure which solves RMQ in $t$ query time, $2n-1.5\log n+n/(\log n)^{O(t\log^2t)}$ bits of space is necessary in the cell-probe model with word-size $\Theta(\log n)$ bits. We emphasize that, in terms of time complexity, our lower bound is tight up to a polylogarithmic factor.