## Abstract

We introduce a simple model illustrating the role of context in communication and the challenge posed by uncertainty of knowledge of context. We consider a variant of distributional communication complexity where Alice gets some information x and Bob gets y, where (x, y) is drawn from a known distribution, and Bob wishes to compute some function g(x, y) (with high probability over (x, y)). In our variant, Alice does not know g, but only knows some function/which is an approximation of g. Thus, the function being computed forms the context for the communication, and knowing it imperfectly models (mild) uncertainty in this context. A naive solution would be for Alice and Bob to first agree on some common function h that is close to both/and g and then use a protocol for h to compute h(x, y). We show that any such agreement leads to a large overhead in communication ruling out such a universal solution. In contrast, we show that if g has a one-way communication protocol with complexity k in the standard setting, then it has a communication protocol with complexity 0(k (1 +/)) in the uncertain setting, where/denotes the mutual information between x and y. In the particular case where the input distribution is a product distribution, the protocol in the uncertain setting only incurs a constant factor blow-up in communication and error. Furthermore, we show that the dependence on the mutual information/is required. Namely, we construct a class of functions along with a non-product distribution over (x, y) for which the communication complexity is a single bit in the standard setting but at least Q, (<Jn) bits in the uncertain setting.

Original language | American English |
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Title of host publication | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 |

Editors | Robert Krauthgamer |

Publisher | Association for Computing Machinery |

Pages | 2072-2085 |

Number of pages | 14 |

ISBN (Electronic) | 9781510819672 |

DOIs | |

State | Published - 2016 |

Externally published | Yes |

Event | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 - Arlington, United States Duration: 10 Jan 2016 → 12 Jan 2016 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 3 |

### Conference

Conference | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 |
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Country/Territory | United States |

City | Arlington |

Period | 10/01/16 → 12/01/16 |

### Bibliographical note

Publisher Copyright:© Copyright (2016) by SIAM: Society for Industrial and Applied Mathematics.