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. 
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37 Citations
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37 Citations

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