随机算法 (Spring 2013)/Problem Set 4: Difference between revisions

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* Give the stationary distribution of the random walk.
* Give the stationary distribution of the random walk.
* Prove that the mixing time is <math>O(n\log k)</math>.
* Prove that the mixing time is <math>O(n\log k)</math>.
== Problem 4 ==
Let <math>G(V,E)</math> be a connected undirected simple graph (no self-loops and parallel edges) defined on <math>n</math> vertices. Let <math>\phi(G)</math> be the expansion ratio of <math>G</math>. <math>G</math> is NOT necessarily regular. For any <math>v\in V</math>, let <math>d_v</math> be the degree of vertex <math>v</math>.
* What is the largest possible value for <math>\phi(G)</math>? Construct a graph <math>G</math> with this expansion ratio and explain why it is the largest.
* What is the smallest possible value for <math>\phi(G)</math>? Construct a graph <math>G</math> with this expansion ratio and explain why it is the smallest.
* Run a lazy random walk on <math>G</math>. What is the stationary distribution? Starting from an arbitrary vertex in an arbitrary unknown <math>G</math>, how long in the worst case should you run the random walk to guarantee the distribution of the current position is within a total variation distance of <math>\epsilon</math> from the stationary distribution? Give an upper bound of the time in terms of <math>n</math> and <math>\epsilon</math>.
* Suppose that the maximum degree of <math>G</math> is known but the graph is not necessarily regular. Design a random walk with uniform stationary distribution. How long should you run the random walk to be within <math>\epsilon</math>-close to the uniform distribution in the worst case?

Revision as of 15:33, 8 June 2013

Problem 1

Consider the Markov chain of graph coloring

Markov Chain for Graph Coloring
Start with a proper coloring of [math]\displaystyle{ G(V,E) }[/math]. At each step:
  1. Pick a vertex [math]\displaystyle{ v\in V }[/math] and a color [math]\displaystyle{ c\in[q] }[/math] uniformly at random.
  2. Change the color of [math]\displaystyle{ v }[/math] to [math]\displaystyle{ c }[/math] if the resulting coloring is proper; do nothing if otherwise.

Show that the Markov chain is:

  1. aperiodic;
  2. irreducible if [math]\displaystyle{ q\ge \Delta+2 }[/math];
  3. with uniform stationary distribution.

Problem 2

Consider the following random walk on hypercube:

Yet another random Walk on Hypercube
At each step, for the current state [math]\displaystyle{ x\in\{0,1\}^n }[/math]:
  1. pick an [math]\displaystyle{ i\in\{0,1,2,\ldots,n\} }[/math] uniformly at random;
  2. flip [math]\displaystyle{ x_i }[/math] (let [math]\displaystyle{ x_i=1-x_i }[/math]) if [math]\displaystyle{ i\neq 0 }[/math].
  • Show that the random walk is ergodic.
  • Give the stationary distribution of the random walk.
  • Analyze the mixing time of the random walk by coupling.

Problem 3

Consider the following random walk over all subsets [math]\displaystyle{ S\in{[n]\choose k} }[/math] for some [math]\displaystyle{ k\le \frac{n}{2} }[/math]:

Random walk over [math]\displaystyle{ k }[/math]-subsets
At each step, for the current state [math]\displaystyle{ S\in{[n]\choose k} }[/math]:
  1. with probability [math]\displaystyle{ p }[/math], do nothing;
  2. else pick an [math]\displaystyle{ x\in S }[/math] and a [math]\displaystyle{ y\in[n]\setminus S }[/math] independently and uniformly at random, and change the current set to be [math]\displaystyle{ S\setminus\{x\}\cup\{y\} }[/math].

You are allowed to choose a self-loop probability [math]\displaystyle{ p }[/math] for your convenience.

  • Show that the random walk is ergodic
  • Give the stationary distribution of the random walk.
  • Prove that the mixing time is [math]\displaystyle{ O(n\log k) }[/math].

Problem 4

Let [math]\displaystyle{ G(V,E) }[/math] be a connected undirected simple graph (no self-loops and parallel edges) defined on [math]\displaystyle{ n }[/math] vertices. Let [math]\displaystyle{ \phi(G) }[/math] be the expansion ratio of [math]\displaystyle{ G }[/math]. [math]\displaystyle{ G }[/math] is NOT necessarily regular. For any [math]\displaystyle{ v\in V }[/math], let [math]\displaystyle{ d_v }[/math] be the degree of vertex [math]\displaystyle{ v }[/math].

  • What is the largest possible value for [math]\displaystyle{ \phi(G) }[/math]? Construct a graph [math]\displaystyle{ G }[/math] with this expansion ratio and explain why it is the largest.
  • What is the smallest possible value for [math]\displaystyle{ \phi(G) }[/math]? Construct a graph [math]\displaystyle{ G }[/math] with this expansion ratio and explain why it is the smallest.
  • Run a lazy random walk on [math]\displaystyle{ G }[/math]. What is the stationary distribution? Starting from an arbitrary vertex in an arbitrary unknown [math]\displaystyle{ G }[/math], how long in the worst case should you run the random walk to guarantee the distribution of the current position is within a total variation distance of [math]\displaystyle{ \epsilon }[/math] from the stationary distribution? Give an upper bound of the time in terms of [math]\displaystyle{ n }[/math] and [math]\displaystyle{ \epsilon }[/math].
  • Suppose that the maximum degree of [math]\displaystyle{ G }[/math] is known but the graph is not necessarily regular. Design a random walk with uniform stationary distribution. How long should you run the random walk to be within [math]\displaystyle{ \epsilon }[/math]-close to the uniform distribution in the worst case?