  Semi-random processes involve an adaptive decision-maker, whose goal is to
achieve some pre-determined objective in an online randomized environment. We
introduce and study a semi-random multigraph process, which forms a
no-replacement variant of the process that was introduced by Ben-Eliezer,
Hefetz, Kronenberg, Parczyk, Shikhelman and Stojakovi\'c (2020). The process
starts with an empty graph on the vertex set $[n]$. For every positive integers
$q$ and $1\leq r\leq n$, in the $((q-1)n+r)$th round of the process, the
decision-maker, called \emph{Builder}, is offered the vertex $\pi_q(r)$, where
$\pi_1, \pi_2, \ldots$ is a sequence of permutations in $S_n$, chosen
independently and uniformly at random. Builder then chooses an additional
vertex (according to a strategy of his choice) and connects it by an edge to
$\pi_q(r)$.
  For several natural graph properties, such as $k$-connectivity, minimum
degree at least $k$, and building a given spanning graph (labeled or
unlabeled), we determine the typical number of rounds Builder needs in order to
construct a graph having the desired property. Along the way we introduce and
analyze an urn model which may also have independent interest.
