# Problem 10: Multi-Round Communication of Gap-Hamming Distance

Consider the communication problem Gap-Hamdist: Alice and Bob are given length $n$ binary strings $x$ and $y$ such that either the Hamming distance $\Delta(x,y) \leq n/2$ or $\Delta(x,y)\geq n/2+\sqrt{n}$. The one-way communication complexity of Gap-Hamdist is known to be $\Omega(n)$ [IndykW-03,Woodruff-04]. Recently, a simpler proof was discovered using a reduction from Index [JayramKS-07]. Is the multi-round communication complexity also $\Omega(n)$? There is a $\Omega(\sqrt{n})$ lower-bound from a reduction from Set-Disjointness but we conjecture that the lower-bound is actually $\Omega(n)$.
If the conjecture is true then it would imply stronger multiple-pass lower bounds for estimating $F_0$ [IndykW-03,Woodruff-04,BarYossefJKST-02] and entropy [BhuvanagiriG-06,ChakrabartiCM-07]. Alternatively, if the conjecture is not true then it would be interesting to see if better multi-pass algorithms exist for $F_0$ and entropy.
This conjecture was proved by Chakrabarti and Regev [ChakrabartiR-11], who showed that the communication complexity of Gap-Hamdist is $\Omega(n)$.