ECCC-Report TR14-055https://eccc.weizmann.ac.il/report/2014/055Comments and Revisions published for TR14-055en-usThu, 11 Aug 2016 22:52:18 +0300
Revision 1
| Communication Complexity of Set-Disjointness for All Probabilities |
Mika Göös,
Thomas Watson
https://eccc.weizmann.ac.il/report/2014/055#revision1We study set-disjointness in a generalized model of randomized two-party communication where the probability of acceptance must be at least alpha(n) on yes-inputs and at most beta(n) on no-inputs, for some functions alpha(n)>beta(n). Our main result is a complete characterization of the private-coin communication complexity of set-disjointness for all functions alpha and beta, and a near-complete characterization for public-coin protocols. In particular, we obtain a simple proof of a theorem of Braverman and Moitra (STOC 2013), who studied the case where alpha=1/2+epsilon(n) and beta=1/2-epsilon(n). The following contributions play a crucial role in our characterization and are interesting in their own right.
(1) We introduce two communication analogues of the classical complexity class that captures small bounded-error computations: we define a "restricted" class SBP (which lies between MA and AM) and an "unrestricted" class USBP. The distinction between them is analogous to the distinction between the well-known communication classes PP and UPP.
(2) We show that the SBP communication complexity is precisely captured by the classical corruption lower bound method. This sharpens a theorem of Klauck (CCC 2003).
(3) We use information complexity arguments to prove a linear lower bound on the USBP complexity of set-disjointness.
Thu, 11 Aug 2016 22:52:18 +0300https://eccc.weizmann.ac.il/report/2014/055#revision1
Paper TR14-055
| Communication Complexity of Set-Disjointness for All Probabilities |
Mika Göös,
Thomas Watson
https://eccc.weizmann.ac.il/report/2014/055We study set-disjointness in a generalized model of randomized two-party communication where the probability of acceptance must be at least alpha(n) on yes-inputs and at most beta(n) on no-inputs, for some functions alpha(n)>beta(n). Our main result is a complete characterization of the private-coin communication complexity of set-disjointness for all functions alpha and beta, and a near-complete characterization for public-coin protocols. In particular, we obtain a simple proof of a theorem of Braverman and Moitra (STOC 2013), who studied the case where alpha=1/2+epsilon(n) and beta=1/2-epsilon(n). The following contributions play a crucial role in our characterization and are interesting in their own right.
(1) We introduce two communication analogues of the classical complexity class that captures small bounded-error computations: we define a "restricted" class SBP (which lies between MA and AM) and an "unrestricted" class USBP. The distinction between them is analogous to the distinction between the well-known communication classes PP and UPP.
(2) We show that the SBP communication complexity is precisely captured by the classical corruption lower bound method. This sharpens a theorem of Klauck (CCC 2003).
(3) We use information complexity arguments to prove a linear lower bound on the USBP complexity of set-disjointness.Thu, 17 Apr 2014 20:49:48 +0300https://eccc.weizmann.ac.il/report/2014/055