In particular, their scheme using exact bounds on energy differences (evaluated using a hierarchical tree as in BH) but in such a way that no approximation is being made. The tree is evaluated to whatever depth is needed to decide whether to accept/reject the MC move (which in worse case could be a brute-force sum over the whole lattice/system) -- this is different I think than BH or other multipole inspired methods (which have a kind of "truncation" or "tolerance" parameter).

[This also works well with systems where update are local and not global, which I think is a difference from some other spatial partitioning schemes -- but I'm less conversant with that aspect].