Monotonicity Testing

Years and Authors of Summarized Original Work

• 1999; Dodis, Goldreich, Lehman, Raskhodnikova, Ron, Samorodnitsky

• 2000; Goldreich, Goldwasser, Lehman, Ron, Samorodnitsky

• 2013; Chakrabarty, Seshadhri

Problem Definition

A real-valued function $f : D \rightarrow \mathbb{R}$ defined over a partially ordered set (poset) D is monotone if f(x) ≤ f(y) for any two points $x \prec y$. In this article, we focus on the poset induced by the coordinates of a d-dimensional, n-hypergrid, [n]^d, where $x \prec y$ iff x _i ≤ y _i for all integers 1 ≤ i ≤ d. Here, we have used [n] as a shorthand for {1, …, n}. The hypercube, {0, 1}^d, and the n-line, [n], are two special cases of this.

Monotonicity testing is the algorithmic problem of deciding whether a given function is monotone. The algorithm has query access to the function, which means that it can query f at any domain point x and obtain the value of f(x). The performance of the algorithm is measured by the number of queries it makes. Although the...