## Abstract

In the coin problem, one is given n independent flips of a coin that has bias β > 0 towards either Head or Tail. The goal is to decide which side the coin is biased towards, with high confidence. An optimal strategy for solving the coin problem is to apply the majority function on the n samples. This simple strategy works as long as β > Ω(1/√n). However, computing majority is an impossible task for several natural computational models, such as bounded width read once branching programs and AC_{0} circuits. Brody and Verbin [8] proved that a length n, width w read once branching program cannot solve the coin problem for β < O(1/(log_{n})_{w}). This result was tightened by Steinberger [20] to O(1/(log n)_{w-2}). The coin problem in the model of AC_{0} circuits was first studied by Shaltiel and Viola [19], and later by Aaronson [1] who proved that a depth d size s Boolean circuit cannot solve the coin problem for β < O(1/(log_{s})_{d+2}). This work has two contributions: We strengthen Steinberger result and show that any Santha-Vazirani source with bias β < O(1/(log n)_{w-2}) fools length n, width w read once branching programs. In other words, the strong independence assumption in the coin problem is completely redundant in the model of read once branching programs, assuming the bias remains small. That is, the exact same result holds for a much more general class of sources. We tighten Aaronson's result and show that a depth d, size s Boolean circuit cannot solve the coin problem for β < O(1/(log_{s})_{d-1}). Moreover, our proof technique is different and we believe that it is simpler and more natural.

Original language | English (US) |
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Title of host publication | Leibniz International Proceedings in Informatics, LIPIcs |

Editors | Klaus Jansen, Cristopher Moore, Nikhil R. Devanur, Jose D. P. Rolim |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 618-629 |

Number of pages | 12 |

ISBN (Electronic) | 9783939897743 |

DOIs | |

State | Published - Sep 1 2014 |

Event | 17th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2014 and the 18th International Workshop on Randomization and Computation, RANDOM 2014 - Barcelona, Spain Duration: Sep 4 2014 → Sep 6 2014 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 28 |

ISSN (Print) | 1868-8969 |

### Other

Other | 17th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2014 and the 18th International Workshop on Randomization and Computation, RANDOM 2014 |
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Country | Spain |

City | Barcelona |

Period | 9/4/14 → 9/6/14 |

## All Science Journal Classification (ASJC) codes

- Software

## Keywords

- Bounded depth circuits
- Read once branching programs
- Santha-Vazirani sources
- The coin problem