  We study the percolation time of the $r$-neighbour bootstrap percolation
model on the discrete torus $(\Z/n\Z)^d$. For $t$ at most a polylog function of
$n$ and initial infection probabilities within certain ranges depending on $t$,
we prove that the percolation time of a random subset of the torus is exactly
equal to $t$ with high probability as $n$ tends to infinity. Our proof rests
crucially on three new extremal theorems that together establish an almost
complete understanding of the geometric behaviour of the $r$-neighbour
bootstrap process in the dense setting. The special case $d-r=0$ of our result
was proved recently by Bollob\'as, Holmgren, Smith and Uzzell.
