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Is t here a Solution?

Is t here a Solution?. Undirected Connectivity. Nondeterministic v s. Random. Random Walks. p 1 (a)=0.5. p 0 (s)=1. Stationary Distribution. Weaker Claim. FixPoint. FixPoint. Using the Asymptotic Estimate. One-Sided Error.

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Is t here a Solution?

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  1. Is there a Solution?

  2. Undirected Connectivity

  3. Nondeterministic vs. Random

  4. Random Walks

  5. p1(a)=0.5 p0(s)=1

  6. Stationary Distribution

  7. Weaker Claim

  8. FixPoint

  9. FixPoint

  10. Using the Asymptotic Estimate

  11. One-Sided Error • 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. Complexity

  12. s t . . . Number of Steps Required? But every time we get here, we get a second chance! The probability we head in the right direction is 1/ds

  13. Complexity How Many Steps Are Needed? • Since expectedly we return to each vertex within|E|/disteps • 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.

  14. Complexity Randomized Algorithm for Undirected Connectivity • Run the random walk from s for 2|V||E| steps. • If node t is ever visited, answer “there is a path from s to t”. • Otherwise, reply “there is probably no path from s to t”.

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

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

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