Lower Bound for Succinct Range Minimum Query

Abstract: Given an integer array $A[1..n]$, the Range Minimum Query problem (RMQ) asks to preprocess $A$ into a data structure, supporting RMQ queries: given $a,b\in [1,n]$, return the index $i\in[a,b]$ that minimizes $A[i]$, i.e., $\mathrm{argmin}_{i\in[a,b]} A[i]$. This problem has a classic solution using $O(n)$ space and $O(1)$ query time by Gabow, Bentley, Tarjan (STOC, 1984) and Harel, Tarjan (SICOMP, 1984). The best known data structure by Fischer, Heun (SICOMP, 2011) and Navarro, Sadakane (TALG, 2014) uses $2n+n/(\frac{\log n}{t})^t+\tilde{O}(n^{3/4})$ bits and answers queries in $O(t)$ time, assuming the word-size is $w=\Theta(\log n)$. In particular, it uses $2n+n/\mathrm{poly}\log n$ bits of space as long as the query time is a constant. In this paper, we prove the first lower bound for this problem, showing that $2n+n/\mathrm{poly}\log n$ space is necessary for constant query time. In general, we show that if the data structure has query time $O(t)$, then it must use at least $2n+n/(\log n)^{\tilde{O}(t^2)}$ space, in the cell-probe model with word-size $w=\Theta(\log n)$.