2. 2

4. 4 Theorem:Proof: What shall we do for an undirected graph?

5. 5

6. 6

7. 7 p 0 (s)=1 p 1 (a)=1/3

8. 8

9. 9

10. 10

11. 11

12. 12

13. Note that if the right answer is ‘NO’, we clearly answer ‘NO’. Hence, this random walk algorithm has one-sided error. Such algorithms are called “Monte-Carlo” algorithms. Complexi ty 13

14. 14 st... The probability we head in the right direction is 1/d i Every visit here, is another chance!

15. Since expectedly we return to each vertex within 2|E|/d i steps The walk expectedly heads in the right direction within 2|E| steps By linearity of the expectation, it is expected to reach t within d(s,t)  2|E|  2|V|  |E| steps. 15 Complexity

16. 1. Run the random walk from s. 2. Once vertex t is visited, halt and accept --- “there is a path from s to t”. 3. Once ran for 8|V|  |E| steps halt and reject --- “there is probably no path from s to t”. The claim is true –by Markov– with probability ≥¼. 16 Complexity

17. Theorem: The above algorithm - uses logarithmic space - always right for ‘NO’ instances. - errs with probability at most ½ for ‘YES’ instances. 17 Complexity To maintain the current position we only need log|V| space Markov: Pr(X>2E[X])<½ PAP 401-404

18. We explored the undirected connectivity problem. We saw a log-space randomized algorithm for this problem. We used an important technique called random walks. 18 Complexity 