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.
翻译:根据不同整数的阵列 $A [1\ ldots n2] 。 最低查询( RMQ) 问题要求我们从 $A 构建一个数据结构, 支持 RMQ 查询 : 如果有间隔 $a, b\ subseteq[ 1, n] 美元, 返回子数组最小元素的索引 $A[ a\ ldots b] $, 即返回 $\ text{argmin} i[ a, b] 。 基本问题有很长的历史。 教科书用 $( n) 字和 $O(1) 来构建一个数据结构, 加布、 本特利、 塔扬( 1984) 和 哈雷尔、 Tarjan( SICOMP 1984) 的时间可以追溯到 1980年代。 Fischer、 Heun( SICOMP 2011) 和 Navarro、 Sadakane ( TALG) 、 任何已知的平面/ tworkn( t) legn( n) t) legn( nx) distrate( n=n) 数( nxxxx) QQQQQQQQQQQQQQQQQQ) lax) lax) lax) lax) lax) 时间( 时间(, 显示一个数字 。