- The algorithm we saw today was independently due to Chu and Liu (1965), Edmonds (1967) and Bock (1971).
- 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.
- The variant of Prim's algorithm Or had mentioned fails on this example:
- 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.

## Wednesday, September 23, 2009

### Lecture 5 notes

Some notes on today's lecture:

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P.S. In the RAM model, Mendelson, Tarjan, Thorup and Zwick (SODA 04, STACS 04) give a deterministic algorithm for branchings that runs in time O(m log log n), and a randomized algorithm that has expected runtime O(m \sqrt{log log n}).

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