  We introduce a general class of algorithms and supply a number of general
results useful for analysing these algorithms when applied to regular graphs of
large girth. As a result, we can transfer a number of results proved for random
regular graphs into (deterministic) results about all regular graphs with
sufficiently large girth. This is an uncommon direction of transfer of results,
which is usually from the deterministic setting to the random one. In
particular, this approach enables, for the first time, the achievement of
results equivalent to those obtained on random regular graphs by a powerful
class of algorithms which contain prioritised actions. As examples, we obtain
new upper or lower bounds on the size of maximum independent sets, minimum
dominating sets, maximum and minimum bisection, maximum $k$-independent sets,
minimum $k$-dominating sets and minimum connected and weakly-connected
dominating sets in $r$-regular graphs with large girth.
