Self-testing/correcting with applications to numerical problems
Journal of Computer and System Sciences - Special issue: papers from the 22nd ACM symposium on the theory of computing, May 14–16, 1990
Property testing and its connection to learning and approximation
Journal of the ACM (JACM)
Journal of Computer and System Sciences - 30th annual ACM symposium on theory of computing
Monotonicity testing over general poset domains
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Robust Characterizations of Polynomials withApplications to Program Testing
SIAM Journal on Computing
Testing Basic Boolean Formulae
SIAM Journal on Discrete Mathematics
Improved Testing Algorithms for Monotonicity
RANDOM-APPROX '99 Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization Problems: Randomization, Approximation, and Combinatorial Algorithms and Techniques
On the strength of comparisons in property testing
Information and Computation
Tolerant property testing and distance approximation
Journal of Computer and System Sciences
Information theory in property testing and monotonicity testing in higher dimension
Information and Computation
Testing Polynomials over General Fields
SIAM Journal on Computing
Estimating the distance to a monotone function
Random Structures & Algorithms
Testing versus Estimation of Graph Properties
SIAM Journal on Computing
Parallel monotonicity reconstruction
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Distribution-Free Property-Testing
SIAM Journal on Computing
Testing monotonicity over graph products
Random Structures & Algorithms
Property-Preserving Data Reconstruction
Algorithmica
Testing of matrix-poset properties
Combinatorica
Fast approximate PCPs for multidimensional bin-packing problems
Information and Computation
Tolerant locally testable codes
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
Distance approximation in bounded-degree and general sparse graphs
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
Algorithmic and Analysis Techniques in Property Testing
Foundations and Trends® in Theoretical Computer Science
Local property reconstruction and monotonicity
Property testing
Local property reconstruction and monotonicity
Property testing
Hi-index | 0.00 |
In this article we study the problem of approximating the distance of a function f: [n]d → R to monotonicity where [n] = {1,&ldots;,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 k-dimensional hypercube [n]k, where 1≤ k≤ d. The quality of estimation that this reduction provides is linear in ⌈ d/k ⌉ 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 O(dlog | 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 k-dimensional hypercube (for any k≥ 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 O(1).