In any method, examining all possible topologies is very time consuming. This algorithm reduces the time spent searching by first producing a temporary tree, (e.g., an NJ tree when an ME tree is being sought), and then examining all of the topologies that are different from this temporary tree by a topological distance of dT = 2 and 4. If this is repeated many times, and all the topologies previously examined are avoided, one can usually obtain the tree being sought.
For the MP method, the CNI search can start with a tree generated by the random addition of sequences. This process can be repeated multiple times to find the MP tree.
See Nei & Kumar (2000) for details.