  We consider the problem of scattering $n$ robots in a two dimensional
continuous space. As this problem is impossible to solve in a deterministic
manner, all solutions must be probabilistic. We investigate the amount of
randomness (that is, the number of random bits used by the robots) that is
required to achieve scattering. We first prove that $n \log n$ random bits are
necessary to scatter $n$ robots in any setting. Also, we give a sufficient
condition for a scattering algorithm to be random bit optimal. As it turns out
that previous solutions for scattering satisfy our condition, they are hence
proved random bit optimal for the scattering problem. Then, we investigate the
time complexity of scattering when strong multiplicity detection is not
available. We prove that such algorithms cannot converge in constant time in
the general case and in $o(\log \log n)$ rounds for random bits optimal
scattering algorithms. However, we present a family of scattering algorithms
that converge as fast as needed without using multiplicity detection. Also, we
put forward a specific protocol of this family that is random bit optimal ($n
\log n$ random bits are used) and time optimal ($\log \log n$ rounds are used).
This improves the time complexity of previous results in the same setting by a
$\log n$ factor. Aside from characterizing the random bit complexity of mobile
robot scattering, our study also closes its time complexity gap with and
without strong multiplicity detection (that is, $O(1)$ time complexity is only
achievable when strong multiplicity detection is available, and it is possible
to approach it as needed otherwise).
