Expanders, Property Testing and the PCP theorem

Lecture 8: Confining a random walk on an expander is hard

Posted in lectures by HQN on September 23, 2008

Let G be an (n,d,\alpha)-spectral expander, B be a subset of vertices of G of size \beta n. Suppose we uniformly choose a random vertex of G and walk randomly for t steps.

Theorem 1 (Confining a random walk is hard). Let (B,t) be the event that the walk is confined within B the entire time.Then, Prob\left[(B,t)\right] \leq \sqrt\beta (\alpha + (1-\alpha)\beta)^t. In particular, if \alpha, \beta<1 then this confinement probability is exponentially small.

Proof. Let \mathbf P = (p_{ij}) be the “projection into B matrix“, i.e. p_{ii} = 1, \forall i \in B and p_{ij}=0 for all other i,j. Noting that \mathbf P is idempotent, it is not difficult to see that

Prob\left[(B,t)\right] = \|(\mathbf{P\hat A})^t\mathbf{Pu}\|_1 = \|(\mathbf{P\hat AP})^t\mathbf{Pu}\|_1 \leq \sqrt n \|(\mathbf{P\hat AP})^t\mathbf{Pu}\|_2

Thus, to bound Prob\left[(B,t)\right] we need to know how much the matrix \mathbf{M = P\hat AP} shrinks a vector after each multiplication. The same trick we did in the last lecture gives, for any non-zero vector \mathbf y

\frac{\|\mathbf{My}\|}{\|\mathbf y\|} \leq \lambda_1(\mathbf M) = \max_{\mathbf z\neq \mathbf 0} \frac{\mathbf z^T\mathbf{Mz}}{\mathbf z^T\mathbf z}

Now, consider any non-zero vector \mathbf z. Let \mathbf{x = Pz}. Then, \mathbf z^T\mathbf{Mz} = \mathbf z^T\mathbf{P\hat APz} = \langle \mathbf{\hat Ax, x} \rangle. First, express \mathbf x as a linear combination of the orthonormal eigenvectors of \mathbf{\hat A}:

\mathbf x = c_1\mathbf u_1 + c_2 \mathbf u_2 + \cdots c_n \mathbf u_n

Let \mathbf x^{\|} = c_1\mathbf u_1 and \mathbf x^{\perp} = \mathbf{x-x^{\|}}. Then,

\langle \mathbf{\hat Ax, x} \rangle = \langle \mathbf{\hat A x^{\|}} + \mathbf{\hat A x^{\perp}}, \mathbf x^{\|} + \mathbf x^{\perp} \rangle = \|\mathbf x^{\|}\|^2 + \langle \mathbf{\hat A x^\perp, x^\perp} \rangle

The usual trick gives

\langle \mathbf{\hat A x^\perp, x^\perp} \rangle = \|\mathbf x^\perp\|^2 \frac{\langle \mathbf{\hat A x^\perp, x^\perp} \rangle}{\|\mathbf x^\perp\|^2} \leq \hat\lambda \|\mathbf x^\perp\|^2 = \hat\lambda \left(\|\mathbf x\|^2 - \|\mathbf x^{\|}\|^2\right)

By Cauchy-Schwarz, \|\mathbf x^{\|}\|^2 = c_1^2 = \langle \mathbf x, \mathbf u_1\rangle^2 \leq \beta \|\mathbf x\|^2, we conclude that

\langle \mathbf{\hat Ax, x} \rangle \leq \hat\lambda\|\mathbf x\|^2 + (1-\hat\lambda)\|\mathbf x^\|\|^2 \leq (\alpha + (1-\alpha)\beta)\|\mathbf x\|^2 \leq (\alpha + (1-\alpha)\beta)\|\mathbf z\|^2

Consequently, each time we apply \mathbf M to a vector \mathbf y we reduce its length by a ratio of at least (\alpha + (1-\alpha)\beta). There are t applications, and the initial vector \mathbf{Pu} has length \sqrt{\beta/n}. The theorem follows.

Theorem 2 (Saving random bits for RP). A language L is in the class RP if there exists a polynomial time randomized algorithm A satisfying the following conditions:

  • x \in L \Rightarrow Prob(A accepts x) \geq 1/2
  • x \notin L \Rightarrow Prob(A accepts x) = 0

Suppose A uses r random bits. To reduce the error probability to 1/2^k, the easy way is to run the algorithm k times and accept the input if the algorithm accepts it at least once. However, this approach requires kr random bits. By imposing a (strongly explicit) (n,d,\alpha)-spectral expander on the space of random strings of length r, we can obtain the same effect with only r + O(k)\log d random bits. Here, we require (\alpha + (1-\alpha)1/2)<1. A strongly explicit expander is an expander which, given a vertex, there’s a poly-time algorithm computing the neighboring vertices.

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