Approximating the distance to monotonicity in high dimensions

Shahar Fattal, Dana Ron*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

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).

Original languageEnglish
Article number52
JournalACM Transactions on Algorithms
Volume6
Issue number3
DOIs
StatePublished - 1 Jun 2010

Keywords

  • Distance approximation
  • Monotonicity
  • Property testing
  • Sublinear approximation algorithms

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