A Polynomial Lower Bound for Testing Monotonicity

An Optimal Lower Bound for Monotonicity Testing over Hypergrids

For positive integers n,d, the hypergrid [n]d is equipped with the coordinatewise product partial ordering denoted by ≺. A function f : [n]d → N is monotone if ∀x ≺ y, f (x)≤ f (y). A function f is ε-far from monotone if at least an ε fraction of values must be changed to make f monotone. Given a parameter ε , a monotonicity tester must distinguish with high probability a monotone function from...

Adaptive Lower Bound for Testing Monotonicity on the Line

In the property testing model, the task is to distinguish objects possessing some property from the objects that are far from it. One of such properties is monotonicity, when the objects are functions from one poset to another. It is an active area of research. Recently, Pallavoor, Raskhodnikova and Varma (ITCS’17) proposed an ε-tester for monotonicity of a function f : [n] → [r], whose complex...

3 Monotonicity Testing Lower Bound : Wrapping up 3

2 TESTING MONOTONICITY : O ( n ) UPPER BOUND

1 Overview 1.1 Last time • Proper learning for P implies property testing of P (generic, but quite inefficient) • Testing linearity (over GF[2]), i.e. P = {all parities}: (optimal) O 1-query 1-sided non-adaptive tester. • Testing monotonicity (P = {all monotone functions}: an efficient O n-query 1-sided non-adaptive tester.

Optimal Complexity Lower Bound for Polynomial Multiplication

We prove lower bounds of order n logn for both the problem of multiplying polynomials of degree n, and of dividing polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower bounds are optimal up to order of magnitude. The proof uses a recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix multiplication. It reduces the li...