A technique we use repeatedly in the chapter is to define a...

A technique we use repeatedly in the chapter is to define a distance function $d_{t}$ that represent the distance between the two states of our coupling after $t$ steps, and then show that when $d_{t}>0$ there exists a $\beta<1$ such that $$ \mathbf{E}\left[d_{t+1} \mid d_{t}\right] \leq \beta d_{t} $$ (a) Under this condition, give an upper bound for $\tau(\varepsilon)$ in terms of $\beta$ and $d^{*}$, where $d^{*}$ is the maximum distance over all possible pairs of initial states for the coupling. (b) Suppose that instead we have $$ \mathbf{E}\left[d_{t+1} \mid d_{t}\right] \leq d_{t} . $$ Suppose we have the additional conditions that $d_{t+1}$ is one of $d_{t}, d_{t}-1$, or $d_{t}+1$ and that $\operatorname{Pr}\left(d_{r} \neq d_{t+1}\right) \geq \gamma$. Give an upper bound for $\tau(\varepsilon)$ in terms of $\varepsilon, d^{*}$, and $\gamma$. Your answer should by polynomial in $d^{*}$ and $1 / \gamma$. (Hint: Think of $d_{t}$ as being similar to a random walk on the line.) (c) Using (a) and (b), show that the mixing time of the coloring chain of Section 11.5 is polynomial in the number of vertices in the graph and $\ln (1 / \varepsilon)$, even when the number of colors is only $2 \Delta$. (d) By extending the argument of part (b), show that the mixing time of the Markov chain for independent sets given in Section $11.6$ is polynomial in the number of vertices in the graph and $\ln (1 / \varepsilon)$.

A technique we use repeatedly in the chapter is to define a distance function $d_{t}$ that represent the distance between the two states of our coupling after $t$ steps, and then show that when $d_{t}>0$ there exists a $\beta<1$ such that
$$
\mathbf{E}\left[d_{t+1} \mid d_{t}\right] \leq \beta d_{t}
$$
(a) Under this condition, give an upper bound for $\tau(\varepsilon)$ in terms of $\beta$ and $d^{*}$, where $d^{*}$ is the maximum distance over all possible pairs of initial states for the coupling.
(b) Suppose that instead we have
$$
\mathbf{E}\left[d_{t+1} \mid d_{t}\right] \leq d_{t} .
$$
Suppose we have the additional conditions that $d_{t+1}$ is one of $d_{t}, d_{t}-1$, or $d_{t}+1$ and that $\operatorname{Pr}\left(d_{r} \neq d_{t+1}\right) \geq \gamma$. Give an upper bound for $\tau(\varepsilon)$ in terms of $\varepsilon, d^{*}$, and $\gamma$. Your answer should by polynomial in $d^{*}$ and $1 / \gamma$. (Hint: Think of $d_{t}$ as being similar to a random walk on the line.)
(c) Using (a) and (b), show that the mixing time of the coloring chain of Section 11.5 is polynomial in the number of vertices in the graph and $\ln (1 / \varepsilon)$, even when the number of colors is only $2 \Delta$.
(d) By extending the argument of part (b), show that the mixing time of the Markov chain for independent sets given in Section $11.6$ is polynomial in the number of vertices in the graph and $\ln (1 / \varepsilon)$.

Key Concepts

Coupling Technique

Coupling is a method used to analyze the convergence properties of Markov chains by constructing a joint probability space for two copies of the chain. By designing the two processes to eventually meet, or 'couple', one can provide bounds on the mixing time of the chain. This general approach is crucial in proving rapid convergence to the stationary distribution.

Distance Function in Couplings

A distance function measures how far apart two states are in the state space of the Markov chain. In coupling arguments, defining an appropriate distance function allows one to track how quickly the coupled processes converge. It is used to quantify the decay of differences between states over time.

Contraction Inequality

The contraction inequality asserts that the expected distance between two coupled states shrinks by a constant factor at each step, under certain conditions. This inequality is central to proving exponential convergence rates since if the distance contracts by a factor less than one, it guarantees rapid mixing of the chain.

Mixing Time

Mixing time is the number of steps required for a Markov chain to become close to its stationary distribution, typically measured in total variation distance. It is a critical concept in the study of stochastic processes and is directly related to the rate at which the distance (as measured by a chosen metric) between different states decreases.

Random Walk Comparison

In situations where the distance function behaves like a random walk (with bounded step sizes), one can use properties of random walks to provide bounds on the convergence time. This connection allows the derivation of polynomial bounds on mixing times by comparing the process to a well-understood random walk model.

Markov Chain Analysis for Graph Coloring and Independent Sets

For specific applications such as graph coloring and independent sets, these coupling and contraction strategies are adapted to manage local constraints and dependencies. By carefully designing the coupling and applying contraction or random walk arguments, one can show that the mixing time remains polynomial in the size of the graph, even under less restrictive configuration spaces.

Transcript

00:04 In these exercises, we are introducing a technique for constructing a deterministic finite state machine, equivalent to a given deterministic finite state machine, with the least number of states possible.

00:23 So we're supposing that m, which is the quintuplet s -i -f -s -0 -f, is a finite state automaton, and that k is a non -negative integer, and we're letting r -k be the relation on the set s of states of states of, them such that s is related to t if and only if for every input string x with the length of x being less than equal to k f of s x and f of t x are both final states or both not final states and we'll also defy r star to be the relation on the set of states m such that s is related to t, if and only if, for every input string x, regardless of the length, f of s of x, and f of t, x are both final states or both not final states.

01:29 Now, in part a, we're asked to show that for every non -negative integer k, rk is an equivalence relation on s.

01:38 And we say that two states, s and t, are k equivalent if s is related to t.

01:57 So, first of all, let's prove the reflexive property.

02:02 So let s be a state of m, and let x be an arbitrary string of length at most k.

02:36 Well, we have that f of sx, and f of sx, by definition, are both final states or both not final states.

03:01 Therefore, it follows that srks is true.

03:10 So it follows that rk is reflexive.

03:15 It was pretty simple.

03:18 Now, let's prove symmetry.

03:21 So we're going to let a be related to b, and let x be an arbitrary string of length at most k.

03:48 Now, since a is related to b, we have that f of a, of a x and f of b x are both final or non -final therefore it follows that f of b x and f of a x are both final or non -final you just simply switch the position of the two and so it follows that b r k a is true and therefore that r k is symmetric.

04:45 And one more property we need to prove is transitivity.

04:50 So suppose that a is related to b and b is related to c.

05:02 This implies that, well, and we're going to let x be string such that the length of x is less than or equal to k.

05:13 Then we know that since a is related to b, f of a x and f of bx are both final or non -final.

05:40 Since b is related to c, we have that f of b x and f of c x are both final or non -final.

05:57 So, it follows that f of a x, if f of a x is final, then f of bx is going to be final, and therefore f of cx is final, so that f of a x and f of cx are both final, or if f of ax is nonfinal, then f of b x is nonfinal.

06:26 And if f of b x is nonfinal, it follows f of c x is nonfinal, so it follows that f of cx is also non -final.

06:35 So f of ax and f of cx are both final or they're both non -final.

06:44 So it follows that a related to c is true and so it follows that rk is transitive.

06:55 And therefore it follows that rk is an equivalence relation.

07:10 In part b, we're asked to show that r star is an equivalence relation on s.

07:22 And we say that two states, s and t, are star equivalent if s is related to t.

07:34 So first, let's prove symmetry.

07:39 So let s be star equivalent to itself.

07:46 Well, no.

07:48 Let s be a state of the machine m.

07:58 And let x be an arbitrary string.

08:28 And we have the length of x.

08:31 Equal to k.

08:50 Now, it follows that from part a, we have that s is related to s through our k.

09:06 So, it follows that f of s x and f of s x are both final or non -final.

09:53 This is actually not the way we want to go, what we want to do, we know that s is related to s through rk for all k by the previous part.

10:13 So we have that since rk is equal to r -star is equal to the infinite intersection of r -k, we have that s is related to s through r -star, and therefore it follows that r -star.

10:55 Is reflexive.

11:06 Now in let's prove that our star is symmetric.

11:14 So let a be star related to b.

11:23 You want to show that b is star related to a.

11:28 So this implies as we saw earlier that a is k related to b for all k but we saw earlier that our k is so then it follows that b is k related to a for all k.

11:55 And therefore it follows that b is star related to a.

12:02 So it follows that r star is symmetric.

12:11 Finally, that's proved that r star is transitive.

12:15 So let a and b be star related and b and c be star related...

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