The asynchronous push&pull protocol, a randomized distributed algorithm for
spreading a rumour in a graph $G$, works as follows. Independent Poisson clocks
of rate 1 are associated with the vertices of $G$. Initially, one vertex of $G$
knows the rumour. Whenever the clock of a vertex $x$ rings, it calls a random
neighbour $y$: if $x$ knows the rumour and $y$ does not, then $x$ tells $y$ the
rumour (a push operation), and if $x$ does not know the rumour and $y$ knows
it, $y$ tells $x$ the rumour (a pull operation). The average spread time of $G$
is the expected time it takes for all vertices to know the rumour, and the
guaranteed spread time of $G$ is the smallest time $t$ such that with
probability at least $1-1/n$, after time $t$ all vertices know the rumour. The
synchronous variant of this protocol, in which each clock rings precisely at
times $1,2,\dots$, has been studied extensively. We prove the following results
for any $n$-vertex graph: In either version, the average spread time is at most
linear even if only the pull operation is used, and the guaranteed spread time
is within a logarithmic factor of the average spread time, so it is $O(n\log
n)$. In the asynchronous version, both the average and guaranteed spread times
are $\Omega(\log n)$. We give examples of graphs illustrating that these bounds
are best possible up to constant factors. We also prove theoretical
relationships between the guaranteed spread times in the two versions. Firstly,
in all graphs the guaranteed spread time in the asynchronous version is within
an $O(\log n)$ factor of that in the synchronous version, and this is tight.
Next, we find examples of graphs whose asynchronous spread times are
logarithmic, but the synchronous versions are polynomially large. Finally, we
show for any graph that the ratio of the synchronous spread time to the
asynchronous spread time is $O(n^{2/3})$.Comment: 25 page