编码范围多数的最优查询时间
Optimal Query Time for Encoding Range Majority
摘要：
我们重新讨论范围$τ$ -多数问题，它要求我们为固定值$τ\in (0,1/2]$预处理数组$A[1..n]$，以便对于任何查询范围$[i,j]$，我们可以返回每个不同的$τ$ -多数元素在$A$中的位置。$τ$ -多数元素是相对频率至少为$[i,j]$范围内的$τ$的元素：即，频率至少为$τ(j-i+1)$。Belazzougui等人[WADS 2013]提出了一种可以在$O(1/τ)$时间内回答此类查询的数据结构，这是最佳的，但空间可以与$Θ(n \lg n)$位一样多。最近，Navarro和Mrsachan [Micromica 2016]表明，可以使用$O(n \lg (1/τ))$位编码来解决这个问题，该编码在空间方面是最佳的，但查询时间次优。在本文中，我们关闭这一差距，并提出了一个数据结构，占用$O(n \lg (1/τ))$位的空间，并有$O(1/τ)$查询时间。我们还表明，这个空间界是最佳的，即使是弱得多的查询，我们必须决定是否查询范围包含至少一个$τ$多数元素。
英文摘要：
We revisit the range $τ$-majority problem, which asks us to preprocess an array $A[1..n]$ for a fixed value of $τ\in (0,1/2]$, such that for any query range $[i,j]$ we can return a position in $A$ of each distinct $τ$-majority element. A $τ$-majority element is one that has relative frequency at least $τ$ in the range $[i,j]$: i.e., frequency at least $τ(j-i+1)$. Belazzougui et al. [WADS 2013] presented a data structure that can answer such queries in $O(1/τ)$ time, which is optimal, but the space can be as much as $Θ(n \lg n)$ bits. Recently, Navarro and Thankachan [Algorithmica 2016] showed that this problem could be solved using an $O(n \lg (1/τ))$ bit encoding, which is optimal in terms of space, but has suboptimal query time. In this paper, we close this gap and present a data structure that occupies $O(n \lg (1/τ))$ bits of space, and has $O(1/τ)$ query time. We also show that this space bound is optimal, even for the much weaker query in which we must decide whether the query range contains at least one $τ$-majority element.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
