Finding optimum branchings

Chu and Liu, Ecbnonds, and Bock have independently devised an ef f ic ient aZgorYithm to f i n d an optimum branching i n a d i rected haph. runs i n O(m2ogn) time i f the problem graph has n vertices and m edges. A modification for dense graphs gives a running time 2 of O(n 1. We also show that the unmodified algorithm runs i n 2 O(n(Zog n) + m) time on an average graph, asswning a uniform probabi Z i t y distribution. We give an impzementation of the algorithm which