Information theory in property testing and monotonicity testing in higher dimension. (English summary) Inform. and Comput. 204 (2006), no. 11, 1704–1717. This paper deals with the question of testing binary functions over the cube [n] d for monotonicity, in the weighted model. Here the weight function is assumed to be a product function, that is, for every x1,..., xn we have D(x1,..., xn) = ∏n i=1 Di(xi) for an appropriate D1,..., Dn. Most of the results relate to the known weight distribution model, i.e., the algorithm knows D1,..., Dn in advance and only needs to query the input function f. The main result is a monotonicity test that uses O ( 2dH ε) many queries, where H is the Shannon entropy of D. This improves upon previous results for large n and concentrated D. The proof works by first analyzing a continuous analogue of the problem for uniform weight distribution, and then showing how this can be transferred back to the discrete model with general produc
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