### Abstract

A function f(x _{1}, ... , 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 ℓ _{1} distance between f and BubbleSmooth(f) is at most twice the ℓ _{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.

Original language | English (US) |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization |

Subtitle of host publication | Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Proceedings |

Pages | 387-398 |

Number of pages | 12 |

DOIs | |

State | Published - Aug 28 2012 |

Event | 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2012 and the 16th International Workshop on Randomization and Computation, RANDOM 2012 - Cambridge, MA, United States Duration: Aug 15 2012 → Aug 17 2012 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7408 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2012 and the 16th International Workshop on Randomization and Computation, RANDOM 2012 |
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Country | United States |

City | Cambridge, MA |

Period | 8/15/12 → 8/17/12 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Proceedings*(pp. 387-398). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7408 LNCS). https://doi.org/10.1007/978-3-642-32512-0_33

}

*Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7408 LNCS, pp. 387-398, 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2012 and the 16th International Workshop on Randomization and Computation, RANDOM 2012, Cambridge, MA, United States, 8/15/12. https://doi.org/10.1007/978-3-642-32512-0_33

**Testing lipschitz functions on hypergrid domains.** / Awasthi, Pranjal; Jha, Madhav; Molinaro, Marco; Raskhodnikova, Sofya.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Testing lipschitz functions on hypergrid domains

AU - Awasthi, Pranjal

AU - Jha, Madhav

AU - Molinaro, Marco

AU - Raskhodnikova, Sofya

PY - 2012/8/28

Y1 - 2012/8/28

N2 - A function f(x 1, ... , 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 ℓ 1 distance between f and BubbleSmooth(f) is at most twice the ℓ 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.

AB - A function f(x 1, ... , 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 ℓ 1 distance between f and BubbleSmooth(f) is at most twice the ℓ 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.

UR - http://www.scopus.com/inward/record.url?scp=84865288626&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84865288626&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-32512-0_33

DO - 10.1007/978-3-642-32512-0_33

M3 - Conference contribution

AN - SCOPUS:84865288626

SN - 9783642325113

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 387

EP - 398

BT - Approximation, Randomization, and Combinatorial Optimization

ER -