随机算法 (Fall 2011)/Problem set 4

Problem 1

Let [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math] be two probability distributions over [math]\displaystyle{ \Omega }[/math]. Give an explicit construction of a coupling [math]\displaystyle{ \mu }[/math] of [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math] such that

[math]\displaystyle{ \Pr_{(X,Y)\sim\mu}[X\neq Y]=\|p-q\|_{TV} }[/math].

Problem 2

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: Pick a vertex [math]\displaystyle{ v\in V }[/math] and a color [math]\displaystyle{ c\in[q] }[/math] uniformly at random. 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 3

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]: pick an [math]\displaystyle{ i\in\{0,1,2,\ldots,n\} }[/math] uniformly at random; 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 4

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]: with probability [math]\displaystyle{ p }[/math], do nothing; 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].