Longest Increasing Subsequence (LIS) is a fundamental statistic of a sequence, and has been studied for decades. While the LIS of a sequence of length $n$ can be computed exactly in time $O(n\log n)$, the complexity of estimating the (length of the) LIS in sublinear time, especially when LIS $\ll n$, is still open. We show that for any integer $n$ and any $\lambda = o(1)$, there exists a (randomized) non-adaptive algorithm that, given a sequence of length $n$ with LIS $\ge \lambda n$, approximates the LIS up to a factor of $1/\lambda^{o(1)}$ in $n^{o(1)} / \lambda$ time. Our algorithm improves upon prior work substantially in terms of both approximation and run-time: (i) we provide the first sub-polynomial approximation for LIS in sub-linear time; and (ii) our run-time complexity essentially matches the trivial sample complexity lower bound of $\Omega(1/\lambda)$, which is required to obtain any non-trivial approximation of the LIS. As part of our solution, we develop two novel ideas which may be of independent interest: First, we define a new Genuine-LIS problem, where each sequence element may either be genuine or corrupted. In this model, the user receives unrestricted access to actual sequence, but does not know apriori which elements are genuine. The goal is to estimate the LIS using genuine elements only, with the minimal number of "genuiness tests". The second idea, Precision Forest, enables accurate estimations for composition of general functions from "coarse" (sub-)estimates. Precision Forest essentially generalizes classical precision sampling, which works only for summations. As a central tool, the Precision Forest is initially pre-processed on a set of samples, which thereafter is repeatedly reused by multiple sub-parts of the algorithm, improving their amortized complexity.
翻译:长期递增子序列( LIS) 是一个序列的基本统计, 并且已经研究了几十年。 虽然一个长度为$n美元序列的LIS 可以精确地按时间计算 $O( n\ log n) 美元, 在亚线时间估计 LIS ( lis) 的长度( lis), 特别是当 LIS $\ll n 美元, 仍然开放时, 我们的算法在任何整数美元和任何美元=lambda = o(1) 美元时, 存在一个( 随机化的) 不适应的算法。 鉴于一个长度为美元( 随机化的) 美元序列, 直线性 美元( 美元/ lambda n) 的序列可以精确地计算 美元, 在亚线下时间序列里程中, 直线性( 直线性) 直线性( 直线性) 直线性( 直线性) 直线性( 直线性) 直线性( 直线性) 直线性( 直线性) 直线性( 直线性) 直线( 直线性) 直线性( 直线性) 直线性( 直线性) 直线性( 直线性) 直线性) 直线性( 直线性) 直线性) 直线性) 函数( 直线性) 直线性) 函数( 等( 直 直 直 直 直 直 直 直 直 直 等( 等) ( 直 直 直 ) ( 直 直 ) 直 直 直 直 直 直 直) ( ) ( ) ( ) ( ) ( ) ( ) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 直 直 ) ( ) ) ( ) ( ) ( ) ) ( ) ) ( ) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (