DeepAI AI Chat
Log In Sign Up

Round Complexity of Common Randomness Generation: The Amortized Setting

by   Noah Golowich, et al.

We study the effect of rounds of interaction on the common randomness generation (CRG) problem. In the CRG problem, two parties, Alice and Bob, receive samples $X_i$ and $Y_i$, respectively, drawn jointly from a source distribution $\mu$. The two parties wish to agree on a common random key consisting of many bits of randomness, by exchanging messages that depend on each party's input and the previous messages. In this work we study the amortized version of the problem, i.e., the number of bits of communication needed per random bit output by Alice and Bob, in the limit as the number of bits generated tends to infinity. The amortized version of the CRG problem has been extensively studied, though very little was known about the effect of interaction on this problem. Recently Bafna et al. (SODA 2019) considered the non-amortized version of the problem: they gave a family of sources $\mu_{r,n}$ parameterized by $r,n\in\mathbb{N}$, such that with $r+2$ rounds of communication one can generate $n$ bits of common randomness with this source with $O(r\log n)$ communication, whereas with roughly $r/2$ rounds the communication complexity is $\Omega(n/{\rm poly}\log n)$. Note that their source is designed with the target number of bits in mind and hence the result does not apply to the amortized setting. In this work we strengthen the work of Bafna et al. in two ways: First we show that the results extend to the amortized setting. We also reduce the gap between the round complexity in the upper and lower bounds to an additive constant. Specifically we show that for every pair $r,n \in \mathbb{N}$ the (amortized) communication complexity to generate $\Omega(n)$ bits of common randomness from the source $\mu_{r,n}$ using $r+2$ rounds of communication is $O(r\log n)$ whereas the amortized communication required to generate the same amount of randomness from $r$ rounds is $\Omega(\sqrt n)$.


page 1

page 2

page 3

page 4


Communication-Rounds Tradeoffs for Common Randomness and Secret Key Generation

We study the role of interaction in the Common Randomness Generation (CR...

Compressed Communication Complexity of Longest Common Prefixes

We consider the communication complexity of fundamental longest common p...

The Price of Uncertain Priors in Source Coding

We consider the problem of one-way communication when the recipient does...

Resource-Efficient Common Randomness and Secret-Key Schemes

We study common randomness where two parties have access to i.i.d. sampl...

Round and Communication Balanced Protocols for Oblivious Evaluation of Finite State Machines

We propose protocols for obliviously evaluating finite-state machines, i...

Learning without Interaction Requires Separation

One of the key resources in large-scale learning systems is the number o...

Efficient Asynchronous Byzantine Agreement without Private Setups

For asynchronous binary agreement (ABA) with optimal resilience, prior p...