The Gap-Hamming-Distance problem arose in the context of proving space
lower bounds for a number of key problems in the data stream model. In
this problem, Alice and Bob have to decide whether the Hamming distance
between their $n$-bit input strings is large (i.e., at least $n/2 +
\sqrt n$) or small (i.e., at most $n/2 - \sqrt n$); they do not care if
it is neither large nor small. This $\Theta(\sqrt n)$ gap in the problem
specification is crucial for capturing the approximation allowed to a
data stream algorithm.

Thus far, for randomized communication, an $\Omega(n)$ lower bound on
this problem was known only in the one-way setting. We prove an
$\Omega(n)$ lower bound for randomized protocols that use any constant
number of rounds.

As a consequence we conclude, for instance, that $\epsilon$-approximately
counting the number of distinct elements in a data stream requires
$\Omega(1/\epsilon^2)$ space, even with multiple (a constant number of)
passes over the input stream. This extends earlier one-pass lower
bounds, answering a long-standing open question. We obtain similar
results for approximating the frequency moments and for approximating
the empirical entropy of a data stream.

In the process, we also obtain tight $n - \Theta(\sqrt{n}\log n)$ lower
and upper bounds on the one-way deterministic communication complexity
of the problem. Finally, we give a simple combinatorial proof of an
$\Omega(n)$ lower bound on the one-way randomized communication
complexity.