We consider monotonicity testing of functions f:[n] d → {0,1}, in the property testing framework of Rubinfeld and Sudan [23] and Goldreich, Goldwasser and Ron [14]. Specifically, we consider the framework of distribution-free property testing, where the distance between functions is measured with respect to a fixed but unknown distribution D on the domain and the testing algorithms have an oracle access to random sampling from the domain according to this distribution D. We show that, though in the uniform distribution case, testing of boolean functions defined over the boolean hypercube can be done using query complexity that is polynomial in \(\frac{1}{\epsilon}\) and in the dimension d, in the distribution-free setting such testing requires a number of queries that is exponential in d. Therefore, in the high-dimensional case (in oppose to the low-dimensional case), the gap between the query complexity for the uniform and the distribution-free settings is exponential.


Boolean Function Monotone Function Query Complexity Property Testing Membership Query 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Shirley Halevy
    • 1
  • Eyal Kushilevitz
    • 1
  1. 1.Department of Computer ScienceTechnionHaifaIsrael

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