  Data dissemination is a fundamental task in distributed computing. This paper
studies broadcast problems in various innovative models where the communication
network connecting $n$ processes is dynamic (e.g., due to mobility or failures)
and controlled by an adversary.
  In the first model, the processes transitively communicate their ids in
synchronous rounds along a rooted tree given in each round by the adversary
whose goal is to maximize the number of rounds until at least one id is known
by all processes.
  Previous research has shown a $\lceil{\frac{3n-1}{2}}\rceil-2$ lower bound
and an $O(n\log\log n)$ upper bound. We show the first linear upper bound for
this problem, namely $\lceil{(1 + \sqrt 2) n-1}\rceil \approx 2.4n$.
  We extend these results to the setting where the adversary gives in each
round $k$-disjoint forests and their goal is to maximize the number of rounds
until there is a set of $k$ ids such that each process knows of at least one of
them. We give a $\left\lceil{\frac{3(n-k)}{2}}\right\rceil-1$ lower bound and a
$\frac{\pi^2+6}{6}n+1 \approx 2.6n$ upper bound for this problem.
  Finally, we study the setting where the adversary gives in each round a
directed graph with $k$ roots and their goal is to maximize the number of
rounds until there exist $k$ ids that are known by all processes. We give a
$\left\lceil{\frac{3(n-3k)}{2}}\right\rceil+2$ lower bound and a $\lceil {
(1+\sqrt{2})n}\rceil+k-1 \approx 2.4n+k$ upper bound for this problem.
  For the two latter problems no upper or lower bounds were previously known.
