# An Optimal Lower Bound for Monotonicity Testing over Hypergrids

@article{Chakrabarty2013AnOL, title={An Optimal Lower Bound for Monotonicity Testing over Hypergrids}, author={Deeparnab Chakrabarty and C. Seshadhri}, journal={Electron. Colloquium Comput. Complex.}, year={2013}, volume={20}, pages={62} }

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

## 37 Citations

A o(d) · polylog n Monotonicity Tester for Boolean Functions over the Hypergrid [n]d

- Computer Science, MathematicsSODA
- 2017

The main technical contribution is a Margulis-style isoperimetric result for the augmented hypergrid, and the non-adaptive tester, like previous testers for the hypercube domain, performs directed random walks on this structure.

Domain Reduction for Monotonicity Testing: A o(d) Tester for Boolean Functions on Hypergrids

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2018

The main technical result is a domain reduction theorem for monotonicity, which shows that for k = poly(d/ε), the expected distance of the restriction E[εf̂ ] = Ω(εf ).

New Algorithms and Lower Bounds for Monotonicity Testing

- Computer Science, Mathematics2014 IEEE 55th Annual Symposium on Foundations of Computer Science
- 2014

A new lower bound is proved on the query complexity of any non-adaptive two-sided error algorithm for testing whether an unknown Boolean function f is monotone versus constant-far from monotones and an algorithm is presented that makes O(n<sup>7/8</sup>) poly(1/ε) queries.

On Monotonicity Testing and Boolean Isoperimetric Type Theorems

- Mathematics, Computer Science2015 IEEE 56th Annual Symposium on Foundations of Computer Science
- 2015

An application gives a monotonicity testing algorithm that makes O̅(√n/ε<sup>2</sup>) non-adaptive queries to a function f always accepts a monotone function and rejects a function that is ε-far from being monotones with constant probability.

Approximating the distance to monotonicity of Boolean functions

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2019

A nonadaptive algorithm that, given oracle access to a function f, makes poly (n,1/α) queries and returns an estimate that, with high probability, is an Õ(n) ‐approximation to the distance of f to monotonicity, obtaining an analogous bound for erasure‐resilient (and tolerant) testers.

Finding Monotone Patterns in Sublinear Time

- Computer Science, Mathematics2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
- 2019

It is shown that the non-adaptive query complexity of finding a length-k monotone subsequence of f : [n] → R, assuming that f is ε-far from free of such subsequences, is Θ((log n)^ ⌊log_2k⌋ ).

K-Monotonicity is Not Testable on the Hypercube

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2017

This work disproves the conjecture that k-monotonicity can be tested with poly(n) queries, and shows that even 2-monotone requires an exponential in √ n number of queries.

Testing k-Monotonicity: The Rise and Fall of Boolean Functions

- Mathematics, Computer ScienceTheory Comput.
- 2019

These techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid [n], and draw connections to distribution testing techniques, as well as demonstrating a separation between testing k-monotonicity and testing monotonicity.

Adaptivity is exponentially powerful for testing monotonicity of halfspaces

- Mathematics, Computer ScienceAPPROX-RANDOM
- 2017

It is shown that adaptivity enables an exponential improvement in the query complexity of monotonicity testing for halfspaces, since non-adaptive algorithms are known to require almost $\Omega(n^{1/2})$ queries to test whether an unknown halfspace is monotone versus far from monotones.

Beyond Talagrand functions: new lower bounds for testing monotonicity and unateness

- Computer Science, MathematicsSTOC
- 2017

A lower bound of Ω(n1/3) is proved for the query complexity of any two-sided and adaptive algorithm that tests whether an unknown Boolean function f:{0, 1}n→ {0,1} is monotone versus far from monotones, a natural generalization of monotonicity.

## References

SHOWING 1-10 OF 27 REFERENCES

Optimal bounds for monotonicity and lipschitz testing over hypercubes and hypergrids

- Computer ScienceSTOC '13
- 2013

The problem of monotonicity testing over the hypergrid and its special case, the hypercube, is a classic question in property testing and a general unified proof for O(n/ε) samples suffice for the edge tester is proved.

Monotonicity testing and shortest-path routing on the cube

- Mathematics, Computer ScienceComb.
- 2010

It is shown that for any δ > 0, the n-dimensional hypercube is not n-realizable with shortest paths, while previously it was only known that hypercubes are not 1- realizable with longest paths.

Improved Testing Algorithms for Monotonicity

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 1999

Improved algorithms for testing monotonicity of functions are presented, given the ability to query an unknown function f: Σ n ↦ Ξ, and the test always accepts a monotone f, and rejects f with high probability if it is e-far from being monotones.

Monotonicity testing over general poset domains

- Mathematics, Computer ScienceSTOC '02
- 2002

It is shown that in its most general setting, testing that Boolean functions are close to monotone is equivalent, with respect to the number of required queries, to several other testing problems in logic and graph theory.

Testing Monotonicity

- MathematicsProceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
- 1998

The analysis of the algorithm relates two natural combinatorial quantities that can be measured with respect to a Boolean function; one being global to the function and the other being local to it.

Fast approximate PCPs for multidimensional bin-packing problems

- Computer Science, MathematicsInf. Comput.
- 1999

Information Theory in Property Testing and Monotonicity Testing in Higher Dimension

- Mathematics, Computer ScienceSTACS
- 2005

On Disjoint Chains of Subsets

- MathematicsJ. Comb. Theory, Ser. A
- 2001

The following theorem concerning the poset of all subsets of n] ordered by inclusion is proved: there exist |S| disjoint saturated chains containing all the subsets in S and R.

Property Testing Lower Bounds via Communication Complexity

- Computer Science, Mathematics2011 IEEE 26th Annual Conference on Computational Complexity
- 2011

A new technique for proving lower bounds in property testing is developed, by showing a strong connection between testing and communication complexity, and significantly strengthens the best known bounds.