TY - JOUR

T1 - Approximating the distance to monotonicity in high dimensions

AU - Fattal, Shahar

AU - Ron, Dana

PY - 2010/6/1

Y1 - 2010/6/1

N2 - In this article we study the problem of approximating the distance of a function Ÿ: [n]d → R to monotonicity where [n] = {1, ⋯, n} and R is some fully ordered range. Namely, we are interested in randomized sublinear algorithms that approximate the Hamming distance between a given function and the closest monotone function. We allow both an additive error, parameterized by δ, and a multiplicative error. Previous work on distance approximation to monotonicity focused on the one-dimensional case and the only explicit extension to higher dimensions was with a multiplicative approximation factor exponential in the dimension d. Building on Goldreich et al. [2000] and Dodis et al. [1999], in which there are better implicit results for the case n = 2, we describe a reduction from the case of functions over the d-dimensional hypercube [n]d to the case of functions over the κ-dimensional hypercube [n]κ, where 1 ≤ κ ≤ d. The quality of estimation that this reduction provides is linear in [d/κ] and logarithmic in the size of the range [R] (if the range is infinite or just very large, then log [R] can be replaced by d log n). Using this reduction and a known distance approximation algorithm for the one-dimensional case, we obtain a distance approximation algorithm for functions over the d-dimensional hypercube, with any range R, which has a multiplicative approximation factor of 0(d log [R]). For the case of a binary range, we present algorithms for distance approximation to monotonicity of functions over one dimension, two dimensions, and the κ-dimensional hypercube (for any κ ≥ 1). Applying these algorithms and the reduction described before, we obtain a variety of distance approximation algorithms for Boolean functions over the d-dimensional hypercube which suggest a trade-off between quality of estimation and efficiency of computation. In particular, the multiplicative error ranges between O(d) and 0(1).

AB - In this article we study the problem of approximating the distance of a function Ÿ: [n]d → R to monotonicity where [n] = {1, ⋯, n} and R is some fully ordered range. Namely, we are interested in randomized sublinear algorithms that approximate the Hamming distance between a given function and the closest monotone function. We allow both an additive error, parameterized by δ, and a multiplicative error. Previous work on distance approximation to monotonicity focused on the one-dimensional case and the only explicit extension to higher dimensions was with a multiplicative approximation factor exponential in the dimension d. Building on Goldreich et al. [2000] and Dodis et al. [1999], in which there are better implicit results for the case n = 2, we describe a reduction from the case of functions over the d-dimensional hypercube [n]d to the case of functions over the κ-dimensional hypercube [n]κ, where 1 ≤ κ ≤ d. The quality of estimation that this reduction provides is linear in [d/κ] and logarithmic in the size of the range [R] (if the range is infinite or just very large, then log [R] can be replaced by d log n). Using this reduction and a known distance approximation algorithm for the one-dimensional case, we obtain a distance approximation algorithm for functions over the d-dimensional hypercube, with any range R, which has a multiplicative approximation factor of 0(d log [R]). For the case of a binary range, we present algorithms for distance approximation to monotonicity of functions over one dimension, two dimensions, and the κ-dimensional hypercube (for any κ ≥ 1). Applying these algorithms and the reduction described before, we obtain a variety of distance approximation algorithms for Boolean functions over the d-dimensional hypercube which suggest a trade-off between quality of estimation and efficiency of computation. In particular, the multiplicative error ranges between O(d) and 0(1).

KW - Distance approximation

KW - Monotonicity

KW - Property testing

KW - Sublinear approximation algorithms

UR - http://www.scopus.com/inward/record.url?scp=77954393623&partnerID=8YFLogxK

U2 - 10.1145/1798596.1798605

DO - 10.1145/1798596.1798605

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AN - SCOPUS:77954393623

SN - 1549-6325

VL - 6

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

IS - 3

M1 - 52

ER -