  The distinct elements problem is one of the fundamental problems in streaming
algorithms --- given a stream of integers in the range $\{1,\ldots,n\}$, we
wish to provide a $(1+\varepsilon)$ approximation to the number of distinct
elements in the input. After a long line of research an optimal solution for
this problem with constant probability of success, using
$\mathcal{O}(\frac{1}{\varepsilon^2}+\log n)$ bits of space, was given by Kane,
Nelson and Woodruff in 2010.
  The standard approach used in order to achieve low failure probability
$\delta$ is to take the median of $\log \delta^{-1}$ parallel repetitions of
the original algorithm. We show that such a multiplicative space blow-up is
unnecessary: we provide an optimal algorithm using $\mathcal{O}(\frac{\log
\delta^{-1}}{\varepsilon^2} + \log n)$ bits of space --- matching known lower
bounds for this problem. That is, the $\log\delta^{-1}$ factor does not
multiply the $\log n$ term. This settles completely the space complexity of the
distinct elements problem with respect to all standard parameters.
  We consider also the \emph{strong tracking} (or \emph{continuous monitoring})
variant of the distinct elements problem, where we want an algorithm which
provides an approximation of the number of distinct elements seen so far, at
all times of the stream. We show that this variant can be solved using
$\mathcal{O}(\frac{\log \log n + \log \delta^{-1}}{\varepsilon^2} + \log n)$
bits of space, which we show to be optimal.
