Presentation is loading. Please wait.

Presentation is loading. Please wait.

2 4 Theorem:Proof: What shall we do for an undirected graph?

Similar presentations


Presentation on theme: "2 4 Theorem:Proof: What shall we do for an undirected graph?"— Presentation transcript:

1

2 2

3

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)=0.5

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 s But every time we get here, we get a second chance!

15 Since expectedly we return to each vertex within|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 for 2|V|  |E| steps. 2. If node t is ever visited, answer “there is a path from s to t”. 3. Otherwise, reply “there is probably no path from s to t”. 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

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 


Download ppt "2 4 Theorem:Proof: What shall we do for an undirected graph?"

Similar presentations


Ads by Google