  We consider the problem of computing a relational query $q$ on a large input
database of size $n$, using a large number $p$ of servers. The computation is
performed in rounds, and each server can receive only $O(n/p^{1-\varepsilon})$
bits of data, where $\varepsilon \in [0,1]$ is a parameter that controls
replication. We examine how many global communication steps are needed to
compute $q$. We establish both lower and upper bounds, in two settings. For a
single round of communication, we give lower bounds in the strongest possible
model, where arbitrary bits may be exchanged; we show that any algorithm
requires $\varepsilon \geq 1-1/\tau^*$, where $\tau^*$ is the fractional vertex
cover of the hypergraph of $q$. We also give an algorithm that matches the
lower bound for a specific class of databases. For multiple rounds of
communication, we present lower bounds in a model where routing decisions for a
tuple are tuple-based. We show that for the class of tree-like queries there
exists a tradeoff between the number of rounds and the space exponent
$\varepsilon$. The lower bounds for multiple rounds are the first of their
kind. Our results also imply that transitive closure cannot be computed in O(1)
rounds of communication.
