Improved Testing Algorithms for Monotonicity

Dodis, Yevgeniy; Goldreich, Oded; Lehman, Eric; Raskhodnikova, Sofya; Ron, Dana; Samorodnitsky, Alex

doi:10.1007/978-3-540-48413-4_10

Cited by 113 publications

(153 citation statements)

References 12 publications

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“…The related notions of tolerant testing and distance approximation were introduced in [26]. The problem of monotonicity in the context of property testing has been studied in [1,10,11,14,15,18,19,22,24]. Sublinear algorithms for approximating the distance of a function to monotonicity have been given in [2,16,26].…”

Section: Our Resultsmentioning

confidence: 99%

“…Specifically we show the following for some constant 0 < α < 1: given a filter on the boolean hypercube {0, 1} d that answers queries within time 2 αd , there is an input function f such that the filter applied to f has error blow-up 2 αd with probability close to 1/2. This shows a complexity gap between testing and reconstruction for the hypercube, since there are monotonicity testers with only a polynomial dependence on d [14,16,22].…”

Section: Our Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Local Property Reconstruction and Monotonicity

Saks

Seshadhri

2010

Property Testing

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We propose a general model of local property reconstruction. Suppose we have a function f on domain Γ, which is supposed to have a particular property P, but may not have the property. We would like a procedure which produces a function g that has property P and is close to f (according to some suitable metric). The reconstruction procedure, called a filter, has the following form. The procedure takes as input an element x of Γ and outputs g(x). The procedure has oracle access to the function f and uses a single short random string ρ, but is otherwise deterministic.This model was inspired by a related model of online property reconstruction that was introduced by by Ailon, Chazelle, . It is related to the property testing model, and extends the framework that is used in the model of locally decodable codes. A similar model, in the context of hypergraph properties, was independently proposed and studied by Austin and Tao (2008).We specifically consider the property of monotonicity and develop an efficient local filter for thie property. The input f is a real valued function defined on the domain {1, . . . , n} d (where n is viewed as large and d as a constant). The function is monotone if the following property holds: for two domain elements x and y, if x ≤ y (in the product order) then f (x) ≤ f (y). Given x, our filter outputs the value g(x) in (log n) O(1) time and uses a random seed ρ of the same size. With high probability, the ratio of the (Hamming) distance between g and f to the minimum possible Hamming distance between a monotone function and f is bounded above by a function of d (independent of n). * This is an extended abstract of work that will appear as "Local Monotonicity Reconstruction" in SIAM Journal of Computing. A preliminary version of this work appeared as "Parallel Monotonicity Reconstruction" [29].

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Section: Our Resultsmentioning

confidence: 99%

Section: Our Resultsmentioning

confidence: 99%

Local Property Reconstruction and Monotonicity

Saks

Seshadhri

2010

Property Testing

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“…Fix i and j. First, we give a standard argument [4,3,7] that it is enough to prove this statement for n × n grids. Namely, every α ∈ [n]…”

Section: Definition 31 (Dimension Operatormentioning

confidence: 98%

“…Such procedures have been previously designed for restoring monotonicity of Boolean functions [4,3] and for restoring the Lipschitz property of functions on the hypercube domain [7]. The key challenge is to find a smoothing procedure that satisfies the following three requirements: (1) It makes all lines along dimension i (i.e., in L Smoothing Procedure for 1-dimensional Functions.…”

Section: Theorem 12 (Dimension Reduction) For All Functions F : [N]mentioning

confidence: 99%

Testing Lipschitz Functions on Hypergrid Domains

et al. 2015

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Abstract. A function f (x1, ..., x d ), where each input is an integer from 1 to n and output is a real number, is Lipschitz if changing one of the inputs by 1 changes the output by at most 1. In other words, Lipschitz functions are not very sensitive to small changes in the input. Our main result is an efficient tester for the Lipschitz property of functions f : [n] d → δZ, where δ ∈ (0, 1] and δZ is the set of integer multiples of δ. The main tool in the analysis of our tester is a smoothing procedure that makes a function Lipschitz by modifying it at a few points. Its analysis is already nontrivial for the 1-dimensional version, which we call Bubble Smooth, in analogy to Bubble Sort. In one step, Bubble Smooth modifies two values that violate the Lipschitz property, i.e., differ by more than 1, by transferring δ units from the larger to the smaller. We define a transfer graph to keep track of the transfers, and use it to show that the 1 distance between f and BubbleSmooth(f ) is at most twice the 1 distance from f to the nearest Lipschitz function. Bubble Smooth has other important properties, which allow us to obtain a dimension reduction, i.e., a reduction from testing functions on multidimensional domains to testing functions on the 1-dimensional domain, that incurs only a small multiplicative overhead in the running time and thus avoids the exponential dependence on the dimension.

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“…The two distance measures we discussed, dist and dist 1 , are identical for arrays with 0/1 entries, which we call Boolean arrays. The L 1 -tester in [2] builds on the sortedness tester for Boolean arrays by Dodis et al [6].…”

Section: Bibliographical Notesmentioning

confidence: 99%