## Abstract

A linear array network consists of k + 1 processors P_{0}, P_{1},..., P_{k} with links only between Pi and P_{i+i} (0 ≤ i < k). It is required to compute some boolean function f(x, y) in this network, where x is initially stored at P_{0} and y at P_{k}. Let D_{k}(f) be the (total) number of bits that must be exchanged to compute/in worst case. Clearly, D_{k}(f) ≤ k · D(f), where D(f) is the standard two-party communication complexity of/. Tiwari proved that for almost all functions D_{k}(f) ≥ k(D(f) - 0(1)) and conjectured that this is true for all functions. In this paper we disprove Tiwari's conjecture, by exhibiting an infinite family of functions for which D_{k}(f) is essentially at most 3/4k · D(f). This construction also leads to progress on another major problem in this area: It is easy to bound the two-party communication complexity of any function, given the least number of monochromatic rectangles in any partition of the input space. How tight are such bounds? We exhibit certain functions, for which the (two-party) communication complexity is twice larger than the best lower bound obtainable this way.

Original language | American English |
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Title of host publication | Proceedings of the 28th Annual ACM Symposium on Theory of Computing, STOC 1996 |

Publisher | Association for Computing Machinery |

Pages | 1-10 |

Number of pages | 10 |

ISBN (Electronic) | 0897917855 |

DOIs | |

State | Published - 1 Jul 1996 |

Event | 28th Annual ACM Symposium on Theory of Computing, STOC 1996 - Philadelphia, United States Duration: 22 May 1996 → 24 May 1996 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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Volume | Part F129452 |

ISSN (Print) | 0737-8017 |

### Conference

Conference | 28th Annual ACM Symposium on Theory of Computing, STOC 1996 |
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Country/Territory | United States |

City | Philadelphia |

Period | 22/05/96 → 24/05/96 |

### Bibliographical note

Funding Information:A linear array network consists of k + 1 processors Po, Pi,..., Pk with links only between Pi and Pi+l (O s i < k). It is required to compute some boolean function ~(~, y) in this network, where z is initially stored at P. and y at Jk, Let Dk(~) be the (total) number of bits that must be exchanged to compute ~ in worst case. Clearly, 11~(~) < k . D(f), where D(f) is the standard two-party communication complexity off. Tiwari proved that for almost all functions Dk(f) > /t(D(f) – O(l)) and conjectured that this is true for all functions. In this paper we disprove Tiwari’s conjecture, by exhibiting an infinite family of functions for which Ilk(f) is essentially at most *Part of this research was done while the authors were at ICSI, Berkeley. t I)epartment of Computer Science, Technion, Israel. http: //www.cs.technion. ac.il/~eyalk. e-mail: eyalk~cs.technion. ac.il. ‘Institute of Computer Science, Hebrew University. Supported in part by a grant from the Israeli Academy of Sciences. e-mail: nati@cs.huji. ac.il $TVork done in its entirety while at ICSI, Berkeley and U.C. Berkeley. Current addr: Bell Communications Research, MCC lG-341R, 445 South Street, Morristown, New Jersey 07960-6438, e-mail: rs&il@bellcore .com

Publisher Copyright:

© 1996 ACM.