Yuval's birthday attack
Yuval's attack is slightly different from the standard birthday attack where we look for a repeated output in a single family of inputs. Instead we look for a repeated output across two families of inputs with at least one member of each family producing the repeated ouput. The probabilities are slightly different, but in a complexity sense are both $O(\sqrt H)$ for output ranges of size $H$ (or $O(2^{m/2})$ for $m$-bit output values) where in the worst case outputs are approximately uniformly distributed.
One can see this in an approximate sense as generating $n$-outputs in each family creates $n^2$ pairs of values with one half of each pair coming from each family. Each pair has a $1/H$ chance of matching the two halves. We can approximate the chance of finding a repeat as
$$1-\left(1-\frac1H\right)^{n^2}$$
and for $n\sim\sqrt H$ this will be about $(1-1/e)$. Note however that this is not a rigorous estimate as the probabilities are not independent. More rigorous derivations are possible (see Girault et al., Eurocrypt '88).