Round Complexity of Common Randomness Generation: The Amortized Setting

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 $Xi$ and $Yi$, 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 $\Omega(\sqrt n)$.