The linear-array conjecture in communication complexity is false

Eyal Kushilevitz; Nathan Linial; Rafail Ostrovsky

doi:10.1145/237814.237817

The linear-array conjecture in communication complexity is false

Eyal Kushilevitz, Nathan Linial, Rafail Ostrovsky

The Rachel and Selim Benin School of Engineering and Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

2 Scopus citations

Abstract

A linear array network consists of k + 1 processors P₀, P₁,..., 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 P₀ 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
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	https://doi.org/10.1145/237814.237817
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
Volume	Part F129452
ISSN (Print)	0737-8017

Conference

Conference	28th Annual ACM Symposium on Theory of Computing, STOC 1996
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.

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10.1145/237814.237817

Cite this

Kushilevitz, E., Linial, N., & Ostrovsky, R. (1996). The linear-array conjecture in communication complexity is false. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, STOC 1996 (pp. 1-10). (Proceedings of the Annual ACM Symposium on Theory of Computing; Vol. Part F129452). Association for Computing Machinery. https://doi.org/10.1145/237814.237817

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title = "The linear-array conjecture in communication complexity is false",

abstract = "A linear array network consists of k + 1 processors P0, P1,..., Pk with links only between Pi and Pi+i (0 ≤ i < k). It is required to compute some boolean function f(x, y) in this network, where x is initially stored at P0 and y at Pk. Let Dk(f) be the (total) number of bits that must be exchanged to compute/in worst case. Clearly, Dk(f) ≤ k · D(f), where D(f) is the standard two-party communication complexity of/. Tiwari proved that for almost all functions Dk(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 Dk(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.",

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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{\textquoteright}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. {\textquoteleft}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: {\textcopyright} 1996 ACM.; 28th Annual ACM Symposium on Theory of Computing, STOC 1996 ; Conference date: 22-05-1996 Through 24-05-1996",

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Kushilevitz, E, Linial, N & Ostrovsky, R 1996, The linear-array conjecture in communication complexity is false. in Proceedings of the 28th Annual ACM Symposium on Theory of Computing, STOC 1996. Proceedings of the Annual ACM Symposium on Theory of Computing, vol. Part F129452, Association for Computing Machinery, pp. 1-10, 28th Annual ACM Symposium on Theory of Computing, STOC 1996, Philadelphia, United States, 22/05/96. https://doi.org/10.1145/237814.237817

The linear-array conjecture in communication complexity is false. / Kushilevitz, Eyal; Linial, Nathan; Ostrovsky, Rafail.
Proceedings of the 28th Annual ACM Symposium on Theory of Computing, STOC 1996. Association for Computing Machinery, 1996. p. 1-10 (Proceedings of the Annual ACM Symposium on Theory of Computing; Vol. Part F129452).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - A linear array network consists of k + 1 processors P0, P1,..., Pk with links only between Pi and Pi+i (0 ≤ i < k). It is required to compute some boolean function f(x, y) in this network, where x is initially stored at P0 and y at Pk. Let Dk(f) be the (total) number of bits that must be exchanged to compute/in worst case. Clearly, Dk(f) ≤ k · D(f), where D(f) is the standard two-party communication complexity of/. Tiwari proved that for almost all functions Dk(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 Dk(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.

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

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The linear-array conjecture in communication complexity is false

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