  We give a time-randomness tradeoff for the quasi-random rumor spreading
protocol proposed by Doerr, Friedrich and Sauerwald [SODA 2008] on complete
graphs. In this protocol, the goal is to spread a piece of information
originating from one vertex throughout the network. Each vertex is assumed to
have a (cyclic) list of its neighbors. Once a vertex is informed by one of its
neighbors, it chooses a position in its list uniformly at random and then
informs its neighbors starting from that position and proceeding in order of
the list. Angelopoulos, Doerr, Huber and Panagiotou [Electron.~J.~Combin.~2009]
showed that after $(1+o(1))(\log_2 n + \ln n)$ rounds, the rumor will have been
broadcasted to all nodes with probability $1 - o(1)$.
  We study the broadcast time when the amount of randomness available at each
node is reduced in natural way. In particular, we prove that if each node can
only make its initial random selection from every $\ell$-th node on its list,
then there exists lists such that $(1-\varepsilon) (\log_2 n + \ln n - \log_2
\ell - \ln \ell)+\ell-1$ steps are needed to inform every vertex with
probability at least $1-O\bigl(\exp\bigl(-\frac{n^\varepsilon}{2\ln
n}\bigr)\bigr)$. This shows that a further reduction of the amount of
randomness used in a simple quasi-random protocol comes at a loss of
efficiency.
