## Tuesday, April 28, 2009

### Lecture 27

In this lecture we covered Reingold's Theorem, SL = L; i.e., undirected connectivity can be solved in log space. Here are my abridged notes on actually doing Reingold's algorithm in log-space. It's not a full proof, just a "proof by example" -- but I hope it's nevertheless clearer than a hand-wave.

For the Martin-Randall Theorem, you can consult the original Martin-Randall paper. There is a clearer proof -- but of a much more general theorem -- in a paper by Jerrum, Son, Tetali, and Vigoda. It's still not the clearest thing. If I get time I will try to write up a stripped-down, simplified version of the proof. You can also consult the Arora-Barak book for a proof of a weaker but still sufficient result: $\epsilon(G r H) \geq \Omega(\epsilon(G) \epsilon(H)^2)$. Note that it is okay to have the square on $\epsilon(H)$ here since it's an absolute constant anyway -- but it would not help if the square were on $\epsilon(G)$.