This paper investigates the standard ordinary least-squares version [24] and the balanced version [20] of the minimum evolution principle. For the standard version, we provide a greedy construction algorithm (GME) which produces a starting topology in O(n ²) time, and a tree swapping algorithm (FASTNNI) that searches for the best topology using nearest neighbor interchanges (NNIs), where the cost of doing p NNIs is O(n ² + pn), i.e. O(n ²) in practice because p is always much smaller than n. The combination of GME and FASTNNI produces trees which are fairly close to Neighbor Joining (NJ) trees in terms of topological accuracy, especially with large trees. We also provide two closely related algorithms for the balanced version, called BME and BNNI, respectively. BME requires O(n ² × diam(T)) operations to build the starting tree, and BNNI is in O(n ² + pn × diam(T)), where diam(T) is the topological diameter of the output tree. In the usual Yule-Harding distribution on phylogenetic trees, the diameter expectation is in log(n), so our algorithms are in practice faster that NJ. Furthermore, BNNI combined with BME (or GME) has better topological accuracy than NJ, BIONJ and WEIGHBOR (all three in O(n ³)), especially with large trees.