The linear-array conjecture in communication complexity is false

Eyal Kushilevitz*, Nathan Linial, Rafail Ostrovsky

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


A linear array network consists of K + 1 processors P0, P1, . . . ,Pk with links only between Pi and Pi+1 (0 ≤ i < k). It is required to compute some boolean function f(x,y) in this network, where initially x is stored at P0 and y is stored at Pk. Let Dk(f) be the (total) number of bits that must be exchanged to compute f in worst case. Clearly, Dk(f)≤k̇D(f), where D(f) is the standard two-party communication complexity of f. Tiwari proved that for almost all functions Dk(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 Dk(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 languageAmerican English
Pages (from-to)241-254
Number of pages14
Issue number2
StatePublished - 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.


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