New Sublinear Algorithms and Lower Bounds for LIS Estimation

10/12/2020
by   Ilan Newman, et al.
0

Estimating the length of the longest increasing subsequence (LIS) in an array is a problem of fundamental importance. Despite the significance of the LIS estimation problem and the amount of attention it has received, there are important aspects of the problem that are not yet fully understood. There are no better lower bounds for LIS estimation than the obvious bounds implied by testing monotonicity (for adaptive or nonadaptive algorithms). In this paper, we give the first nontrivial lower bound on the complexity of LIS estimation, and also provide novel algorithms that complement our lower bound. Specifically, for every constant ϵ∈ (0,1), every nonadaptive algorithm that outputs an estimate of the length of the LIS in an array of length n to within an additive error of ϵ· n has to make log^Ω(log (1/ϵ)) n) queries. Next, we design nonadaptive LIS estimation algorithms whose complexity decreases as the the number of distinct values, r, in the array decreases. We first present a simple algorithm that makes Õ(r/ϵ^3) queries and approximates the LIS length with an additive error bounded by ϵ n. We then use it to construct a nonadaptive algorithm with query complexity Õ(√(r)·poly(1/λ)) that, for an array with LIS length at least λ n, outputs a multiplicative Ω(λ)-approximation to the LIS length. Finally, we describe a nonadaptive erasure-resilient tester for sortedness, with query complexity O(log n). Our result implies that nonadaptive tolerant testing is strictly harder than nonadaptive erasure-resilient testing for the natural property of monotonicity.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

11/04/2019

Optimal Adaptive Detection of Monotone Patterns

We investigate adaptive sublinear algorithms for detecting monotone patt...
11/16/2019

Approximating the Distance to Monotonicity of Boolean Functions

We design a nonadaptive algorithm that, given a Boolean function f{0,1}^...
10/03/2019

Finding monotone patterns in sublinear time

We study the problem of finding monotone subsequences in an array from t...
09/07/2019

Hard properties with (very) short PCPPs and their applications

We show that there exist properties that are maximally hard for testing,...
01/24/2022

Optimal SQ Lower Bounds for Learning Halfspaces with Massart Noise

We give tight statistical query (SQ) lower bounds for learnining halfspa...
10/18/2018

Testing Matrix Rank, Optimally

We show that for the problem of testing if a matrix A ∈ F^n × n has rank...
06/09/2020

Sublinear Algorithms and Lower Bounds for Metric TSP Cost Estimation

We consider the problem of designing sublinear time algorithms for estim...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.