## Abstract

A linear array network consists of K + 1 processors P_{0}, P_{1}, . . . ,P_{k} with links only between P_{i} and P_{i}+1 (0 ≤ i < k). It is required to compute some boolean function f(x,y) in this network, where initially x is stored at P_{0} and y is stored at P_{k}. Let D_{k}(f) be the (total) number of bits that must be exchanged to compute f in worst case. Clearly, D_{k}(f)≤k̇D(f), where D(f) is the standard two-party communication complexity of f. Tiwari proved that for almost all functions D_{k}(f)≥k(D(f)-O(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 3/4k̇ D(f). Our 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 as large as the best lower bound obtainable this way.

Original language | English |
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Pages (from-to) | 241-254 |

Number of pages | 14 |

Journal | Combinatorica |

Volume | 19 |

Issue number | 2 |

DOIs | |

State | Published - 1999 |

### Bibliographical note

Funding Information:Mathematics Subject Classi cation (1991): 68Q22, 68Q10, 94A05 * Part of this research was done while the authors were at ICSI, Berkeley. An early version of this paper appeared in the proceedings of the 28th ACM Symp. on Theory of Computing (STOC), pp. 1{10, May 1996. y Supported in part by a grant from the Israeli Academy of Sciences.