Mazes

1
Mazes

• And Random Walks

2
Can You Solve This Maze?
3
The Solution
4
What Will We Do?

• Our task is to show an algorithm for general
  mazes.
• We have memory which is logarithmic in the size
  of the maze.

5
Introduction

• Objectives
• To explore the undirected connectivity problem
• To introduce randomized computations
• Overview
• Undirected Connectivity
• Random Walks

6
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

7
What Do We Know?

• Theorem
• Directed Connectivity is NL-Complete
• Corollary
• Undirected Connectivity is in NL.

8
Undirected Connectivity is in NL Revisit

• Our non-deterministic algorithm

At each node, non-deterministically choose a
neighbor and jump to it
9
What If We Dont 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

10
Idea!

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

11
Random Walks

• Add a self loop to each vertex.
• Start at s.
• Let di be the degree of the current node.
• Jump to each of the neighbors with probability
  1/di.
• Stop if you get to t.

12
Notations

• Let vt denote the node visited at time t (v0s).
• Let pt(i) Prvti

p1(a)0.5
p0(s)1
13
Stationary Distribution

• Lemma
• If G(V,E) is a connected graph,
• for any i?V,

14
Weaker Claim

• Well prove a weaker result
• Lemma If for some t, for any i?V,
• then for any i?V,

15
Proof

• Proof ?di2E. If the ith node has weight di at
  time t, then it retains this weight at time t1
  (its reachable (only) from its di neighbors). ?

16
Illustrated Proof
17
Using the Asymptotic Estimate

• Corollary Starting from some node i, we will
  revisit i within expectedly 2E/di steps.
• Proof Since the walk has no memory, the
  expected return time is the same as the
  asymptotic estimate?

18
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.

19
How Many Steps Are Needed?

• If the right answer is YES, in how many steps
  do we expect to discover that?

But every time we get here, we get a second
chance!
The probability we head in the right direction is
1/ds
20
How Many Steps Are Needed?

• Since expectedly we return to each vertex after
  2E/di 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.

21
Randomized Algorithm for Undirected Connectivity

1. Run the random walk from s for 2V?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.

22
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
logV space
Markov Pr(Xgt2EX)lt½
23
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.