  We consider the classic Set Cover problem in the data stream model. For $n$
elements and $m$ sets ($m\geq n$) we give a $O(1/\delta)$-pass algorithm with a
strongly sub-linear $\tilde{O}(mn^{\delta})$ space and logarithmic
approximation factor. This yields a significant improvement over the earlier
algorithm of Demaine et al. [DIMV14] that uses exponentially larger number of
passes. We complement this result by showing that the tradeoff between the
number of passes and space exhibited by our algorithm is tight, at least when
the approximation factor is equal to $1$. Specifically, we show that any
algorithm that computes set cover exactly using $({1 \over 2\delta}-1)$ passes
must use $\tilde{\Omega}(mn^{\delta})$ space in the regime of $m=O(n)$.
Furthermore, we consider the problem in the geometric setting where the
elements are points in $\mathbb{R}^2$ and sets are either discs, axis-parallel
rectangles, or fat triangles in the plane, and show that our algorithm (with a
slight modification) uses the optimal $\tilde{O}(n)$ space to find a
logarithmic approximation in $O(1/\delta)$ passes.
  Finally, we show that any randomized one-pass algorithm that distinguishes
between covers of size 2 and 3 must use a linear (i.e., $\Omega(mn)$) amount of
space. This is the first result showing that a randomized, approximate
algorithm cannot achieve a space bound that is sublinear in the input size.
  This indicates that using multiple passes might be necessary in order to
achieve sub-linear space bounds for this problem while guaranteeing small
approximation factors.
