Lecture 5 notes

Some notes on today's lecture:

1. The algorithm we saw today was independently due to Chu and Liu (1965), Edmonds (1967) and Bock (1971).
   
   
2. The best running time for the min-cost rooted arborescence is apparently O(m + n log n) due to Gabow, Galil, Spencer and Tarjan, in the same paper that gave the O(m log \beta(m,n)) algorithm for undirected MSTs.
   
   
3. The variant of Prim's algorithm Or had mentioned fails on this example:
4. I didn't explicitly say why the final two claims proved optimality for the branching F*, here is the reason --- if there were a branching F' that had smaller cost, then by the second claim it would have cost strictly less than \sum_S w*_S. But since w* was a valid weight function, that would violate the first claim.