  The element distinctness problem is the problem of determining whether the
elements of a list are distinct, that is, if $x=(x_1,...,x_N)$ is a list with
$N$ elements, we ask whether the elements of $x$ are distinct or not. The
solution in a classical computer requires $N$ queries because it uses sorting
to check whether there are equal elements. In the quantum case, it is possible
to solve the problem in $O(N^{2/3})$ queries. There is an extension which asks
whether there are $k$ colliding elements, known as element $k$-distinctness
problem.
  This work obtains optimal values of two critical parameters of Ambainis'
seminal quantum algorithm [SIAM J.~Comput., 37, 210-239, 2007]. The first
critical parameter is the number of repetitions of the algorithm's main block,
which inverts the phase of the marked elements and calls a subroutine. The
second parameter is the number of quantum walk steps interlaced by oracle
queries. We show that, when the optimal values of the parameters are used, the
algorithm's success probability is $1-O(N^{1/(k+1)})$, quickly approaching 1.
The specification of the exact running time and success probability is
important in practical applications of this algorithm.
