# Exponential Separation of Information and Communication for Boolean Functions

@article{Ganor2014ExponentialSO, title={Exponential Separation of Information and Communication for Boolean Functions}, author={Anat Ganor and Gillat Kol and Ran Raz}, journal={Proceedings of the forty-seventh annual ACM symposium on Theory of Computing}, year={2014} }

We show an exponential gap between communication complexity and information complexity for boolean functions, by giving an explicit example of a partial function with information complexity ≤ O(k), and distributional communication complexity ≥ 2k. This shows that a communication protocol for a partial boolean function cannot always be compressed to its internal information. By a result of Braverman [Bra12], our gap is the largest possible. By a result of Braverman and Rao [BR11], our example…

## Figures from this paper

## 36 Citations

Exponential separation of communication and external information

- Computer ScienceSTOC
- 2015

An explicit example of a search problem with external information complexity ≤ O(k), withrespect to any input distribution, and distributional communication complexity ≥ 2k, with respect to some input distribution is obtained.

Exponential separation of quantum communication and classical information

- Computer ScienceSTOC
- 2017

A simple proof for an optimal trade-off between Alice's and Bob's communication is given, even when allowing pre-shared entanglement, while computing the related Greater-Than function on n bits.

Quantum Advantage on Information Leakage for Equality

- Computer ScienceArXiv
- 2016

We prove a lower bound on the information leakage of any classical protocol computing the equality function in the simultaneous message passing (SMP) model. Our bound is valid in the finite length…

Relative Discrepancy Does Not Separate Information and Communication Complexity

- MathematicsACM Trans. Comput. Theory
- 2015

It is shown that in the non-distributional setting, the relative discrepancy bound is smaller than the information complexity; hence, it cannot separate information and communication complexity.

Communication Complexity of Set-Disjointness for All Probabilities

- Computer Science, MathematicsAPPROX-RANDOM
- 2014

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.

Information Lower Bounds via Self-Reducibility

- Computer ScienceTheory of Computing Systems
- 2015

We use self-reduction methods to prove strong information lower bounds on two of the most studied functions in the communication complexity literature: Gap Hamming Distance (GHD) and Inner Product…

The Landscape of Communication Complexity Classes

- Computer Science, MathematicsICALP
- 2016

We prove several results which, together with prior work, provide a nearly-complete picture of the relationships among classical communication complexity classes between P and PSPACE, short of…

Multi-Party Protocols, Information Complexity and Privacy

- Computer ScienceACM Trans. Comput. Theory
- 2019

A new information-theoretic measure, which is able to use directly in the natural asynchronous message-passing peer-to-peer model, and a number of interesting properties and applications of this measure are shown.

Multi-Party Protocols, Information Complexity and Privacy

- Computer ScienceMFCS
- 2016

We introduce a new information theoretic measure that we call Public Information Complexity (PIC), as a tool for the study of multi-party computation protocols, and of quantities such as their…

Information Complexity Density and Simulation of Protocols Extended Abstract

- Computer Science
- 2016

In the amortized regime for product protocols, the exact second order term, together with the precise dependence on epsilon, is identified and a general formula for the leading asymptotic term is derived.

## References

SHOWING 1-10 OF 47 REFERENCES

Exponential separation of communication and external information

- Computer ScienceSTOC
- 2015

An explicit example of a search problem with external information complexity ≤ O(k), withrespect to any input distribution, and distributional communication complexity ≥ 2k, with respect to some input distribution is obtained.

Exponential Separation of Information and Communication

- Computer Science2014 IEEE 55th Annual Symposium on Foundations of Computer Science
- 2014

An exponential gap between communication complexity and information complexity is shown, by giving an explicit example for a communication task (relation), with information complexity ≤ O(k), and distributional communication complexity ≥2k, implying that a tight direct sum result for distributional communications complexity cannot hold.

A Direct Sum Theorem in Communication Complexity via Message Compression

- Computer Science, MathematicsICALP
- 2003

The main technical result is a 'compression' theorem saying that, for any probability distribution µ over the inputs, a k-round private coin bounded error protocol for a function f can be converted into aK-round deterministic protocol for f with bounded distributional error and communication cost O(kc).

A Direct Product Theorem for the Two-Party Bounded-Round Public-Coin Communication Complexity

- Computer Science, MathematicsFOCS
- 2012

A strong direct product theorem for a problem in a given model of computation states that, in order to compute k instances of the problem, if the authors provide resource which is less than k times the resource required for computing one instance of the Problem, then the probability of correctly computing all the k instances together, is exponentially small in k.

A strong direct product theorem for disjointness

- Mathematics, Computer ScienceSTOC '10
- 2010

A strong direct product theorem is established that if the authors want to compute k independent instances of a function, using less than k times the resources needed for one instance, then the overall success probability will be exponentially small in k, which solves an open problem of [KSW07, LSS08].

A Direct Product Theorem for Two-Party Bounded-Round Public-Coin Communication Complexity

- Computer Science, Mathematics2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
- 2012

A direct product theorem for the communication complexity of any complete relation in this model of two-party bounded-round public-coin randomized communication complexity is shown and can be considered as an important progress towards settling the strong direct product conjecture for two- party public-coins communication complexity.

New Strong Direct Product Results in Communication Complexity

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2011

It is proved that the new complexity measure gives a tight lower bound of Ω(n) for the set-disjointness problem on n-bit inputs (this strengthens the linear lower bound on the rectangle/corruption bound for set- Disjoints shown by Razborov [1992]).

Information Equals Amortized Communication

- Computer Science, MathematicsIEEE Transactions on Information Theory
- 2011

We show how to efficiently simulate the sending of a single message M to a receiver who has partial information about the message, so that the expected number of bits communicated in the simulation…

Informational complexity and the direct sum problem for simultaneous message complexity

- Computer Science, MathematicsProceedings 2001 IEEE International Conference on Cluster Computing
- 2001

A new notion of informational complexity is introduced which is related to SM complexity and has nice direct sum properties and appears to be quite powerful and may be of independent interest.

Lower Bounds on Information Complexity via Zero-Communication Protocols and Applications

- Computer Science, MathematicsSIAM J. Comput.
- 2012

A relaxed version of the partition bound of Jain and Klauck is defined and it is proved that it lower bounds the information complexity of any function.