1 Mazes In The Theory of Computer Science Dana Moshkovitz

2 Can You Solve This Maze?

3 The Solution

4 Introduction Objectives: –To explore the undirected connectivity problem –To introduce randomized computations Overview: –Undirected Connectivity –Random Walks

5 Undirected Connectivity Instance: An undirected graph G=(V,E) and two vertices s,t  V Problem: To decide if there is a path from s to t in G

6 What Do We Know? Theorem: Directed Connectivity is NL-Complete Corollary: Undirected Connectivity is in NL.

7 Undirected Connectivity is in NL: Revisit Our non-deterministic algorithm: At each node, non- deterministically choose a neighbor and jump to it

8 What If We Don’t Have “Magic Coins”? Non-deterministic “algorithms” use “magic coins” to lead them to the right solution if one exists. In real life, these are unavailable…

9 Idea! What if we have plain coins? In other words, what if we randomly choose a neighbor?

10 Random Walks Add a self loop to each vertex. Start at s. Let d i be the degree of the current node. Jump to each of the neighbors with probability 1/d i. Stop if you get to t. st

11 Notations Let v t denote the node visited at time t (v 0 =s). Let p t (i) = Pr[v t =i] p 0 (s)=1 p 1 (a)=0.5 sta

12 Stationary Distribution Lemma: If G=(V,E) is a connected graph, for any i  V,

13 Weaker Claim We’ll prove a weaker result: Lemma: If for some t, for any i  V, then for any i  V,

14 Proof Proof:  d i =2|E|. If the i th node has weight d i at time t, then it retains this weight at time t+1 (it’s reachable (only) from its d i neighbors). 

15 Illustrated Proof

16 Using the Asymptotic Estimate Corollary: Starting from some node i, we will revisit i within expectedly 2|E|/d i steps.

17 One-Sided Error Note that if the right answer is ‘NO’, we clearly answer ‘NO’. Thus, a random walk algorithm has one- sided error. Such algorithms are called “Monte-Carlo” algorithms.

18 How Many Steps Are Needed? If the right answer is ‘YES’, in how many steps do we expect to discover that? st... The probability we head in the right direction is 1/d s But every time we get here, we get a second chance!

19 How Many Steps Are Needed? Since expectedly we return to each vertex after 2|E|/d i steps, We expect to head in the right direction after |E| steps (w.p. ½). By the linearity of the expectation, we expect to encounter t in d(s,t)  |E|  |V|  |E| steps.

20 Randomized Algorithm for Undirected Connectivity 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”.

21 Main Theorem 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 PAP

22 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. 