Deconstructing 1-Local Expanders

Abstract

A 1-local 2d-regular $2^n$-vertex graph is represented by d bijections over $\{0,1\}^n$ such that each bit in the output of each bijection is a function of a single bit in the input. An explicit construction of 1-local expanders was presented by Viola and Wigderson (ECCC, TR16-129, 2016), and the goal of the current work is to de-construct it; that is, make its underlying ideas more transparent.

Starting from a generic candidate for a 1-local expander (over $\{0,1\}^n$), we first observe that its underlying bijections consists of pairs of (“relocation”) permutations over [n] and offsets (which are n-bit long strings). Next, we formulate a natural problem regarding “coordinated random walks” (CRW) on the corresponding (n-vertex) “relocation” graph, and prove the following two facts:

• 1. Any solution to the CRW problem yields 1-local expanders.

• 2. Any constant-size expanding set of generators for the symmetric group (over [n]) yields a solution to the CRW problem.

This yields an alternative construction and different analysis than the one used by Viola and Wigderson. Furthermore, we show that solving (a relaxed version of) the CRW problem is equivalent to constructing 1-local expanders.

Appendix: Secondary Observations

This appendix contains proofs of two secondary observations that were mentioned in Sect. 1.1. We also include additional evidence that 1-local expanders may be constructed without using an expanding set of generators for $\mathtt{Sym}_n$ (see Theorem A.2).

Improving over Observation 1.2. A natural way of trying to improve over Observation 1.2 is to use relocation graphs that have shorter cover time. The natural choice is to use n-vertex expander graphs.²⁰

Observation A.1

(using an expander for the relocation graph): Let $\pi ^{(1)},...,\pi ^{(d)}:[n]\rightarrow [n]$ be bijections that represent the edges of an 2d-regular expander, and $\mathtt{id}:[n]\rightarrow [n]$ denote the identity bijection. Then, the 1-local $2^n$-vertex graph associated with the 2d relocation permutation $\pi ^{(1)},...,\pi ^{(d)},\mathtt{id},...,\mathtt{id}$ and the 2d offsets $0^n,...,0^n,0^{n-1}1,...,0^{n-1}1$ (i.e,., the $i^\mathrm{th}$ bijection is $x\mapsto x_{\pi ^{(i)}}$ if $i\in [d]$ and $x\mapsto x\oplus 0^{n-1}1$ otherwise) has second (normalized) eigenvalue $1-\varTheta (1/n\log n)$.

Proof Sketch:

In this case, a random walk of length $t=O(t'\cdot n\log n)$ on the n-vertex graph visits all vertices with probability at least $1-2^{-t'}$ (since its cover time is $O(n\log n)$ and we have $t'$ “covering attempts”).²¹ It follows that taking a random walk of length $O(t'\cdot n\log n)$ on the 1-local graph yields a distribution that is $2^{-t'}$-close to uniform, since (with probability $1-2^{-t'}$) each position in the original n-bit string is mapped to the rightmost position at some time, and at the next step the corresponding value is “randomized” (since the offset is applied with probability 1/2).    $\blacksquare$

Proposition 1.4 (revised): Consider a 2d-regular $2^n$-vertex graph as in Definition 1.3, and suppose that for every $i\in [d]$ either $|s^{(i)}|=o(n)$ or $|s^{(i)}|=n-o(n)$. Then, this 1-local $2^n$-vertex graph is not an expander.

Proof:

For starters, we assume that $|s^{(i)}|=o(n)$ for every $i\in [d]$. We first consider an auxiliary 4d-regular $2^n$-vertex graph in which, for each $i\in [d]$, the $i^\mathrm{th}$ relocation permutation (i.e., $\pi ^{(i)}$) is coupled both with the offset $s^{(i)}$ and with the all-zero offset.

The key observation is that, during a random walk on this 1-local $2^n$-vertex graph, bits in the label of the current vertex get randomized by the offsets with too small probability, since at each step only o(n) locations are randomized. Specifically, for a t-step random walk that starts at the vertex $0^n$, consider the event this walk does not randomize position $j\in [n]$ (in the initial n-bit string); that is, the corresponding walk on the n-vertex relocation graph that starts at vertex $j\in [n]$ does not go through any vertex in the set $S{\mathop {=}\limits ^\mathrm{def}}\bigcup _{i\in [d]}\{k:s^{(i)}_k=1\}$. This bad event (which refers to a random walk that starts at j) occurs with probability at least $\eta =\exp (-o(t))/n$, because the probability that a walk of length t that starts at a random vertex on any O(1)-regular n-vertex graph misses a set of o(n) vertices is at least $(1-o(1))^t=\exp (-o(t))$.²²

Note that each randomized bit position is reset to 1 with probability exactly 1/2 (by virtue of the auxiliary construction performed upfront), whereas each non-randomized position maintains the value 0. Considering the expected number of ones in the label of the final vertex of a t-step random walk (on the $2^n$-vertex graph), observe that if some bit is not randomized with probability $\eta$, then the expected number of ones is at most $(1\,-\,\eta )\,\cdot \, 0.5\,\cdot \, n\,+\,\eta \,\cdot \,0.5\,\cdot \,(n\,-\,1) = (n\,-\,\eta )/2$. It follows that the total variation distance between the distribution of the final vertex and the uniform distribution is at least $\eta '{\mathop {=}\limits ^\mathrm{def}}\eta /2n=\exp (-o(t)-\log n)$.²³

We stress that the foregoing holds for any t, which means that we assume that $n=o(t)$, let alone $\log n=o(t)$. Hence, the convergence rate of the 1-local $2^n$-vertex graph is not bounded away from 1 (since $\eta '=\exp (-o(t))$ whereas the convergence rate $\lambda$ must satisfy $2^n\cdot \lambda ^t > \eta '$).²⁴ Lastly, we note that given that the auxiliary graph is not an expander, the original graph (which is a subgraph of it) is also not an expander.

Turning to the case in which also offsets of Hamming weight $n-o(n)$ exist, we note that this is equivalent to using an offset of weight o(n) and complementing all bits in the resulting label. Hence, such offsets can randomize many individual locations but cannot randomize all pairs of locations (i.e., randomize each location independently of its paired location). Hence, we extend the foregoing argument to pairs of locations.

We first observe that there are two positions $j_1\ne j_2$ such that with probability $\eta '=\exp (-o(t))$ these position are always randomized together (i.e., in each steps either both $j_1$ and $j_2$ are in locations that get randomized by some single offset or both are not in such locations).²⁵ The argument is completed by considering the expected number of pairs of positions that hold the same value.

   $\blacksquare$

Added in Revision: Additional Evidence Against the Need for an Expanding Set of Generators for $\mathtt{Sym}_{\varvec{n}}$. Theorem 3.2 asserts that the relocation permutations used in a 1-local $2^n$-vertex expander graph need not generate the symmetric group over [n], let alone in an expanding manner. This indicates that sets of expanding generators for $\mathtt{Sym}_n$ may not be essential for the construction of 1-local expanders. Additional evidence in that direction is provided by the following composition of 1-local graphs.

Theorem A.2

(composing 1-local expanders): Suppose that, for $j\in \{1,2\}$, there is a $2d_j$-regular 1-local $2^{n_j}$-vertex expander graph, which uses the 1-local bijections $f^{(j)}_1,...,f^{(j)}_{d_j}:\{0,1\}^{n_j}\rightarrow \{0,1\}^{n_j}$. Then, the $4d_1d_2$-regular 1-local $2^{n_1+n_2}$-vertex graph that uses the set of 1-local bijections

$$\begin{aligned} \{f_{i_1,i_2,b}:\{0,1\}^{n_1+n_2} \rightarrow \{0,1\}^{n_1+n_2}\}_{i_1\in [d_1],i_2\in [d_2],b\in \{\pm 1\}} \end{aligned}$$

(4)

where $f_{i_1,i_2,b}(yz)=f^{(1)}_{i_1}(y)(f^{(2)}_{i_2})^{b}(z)$, is an expander.

Note that the corresponding set of relocation permutations does not generate the symmetric group of $[n_1+n_2]$, since it generates at most $(n_1!)\cdot (n_2!)$ permutations. We comment that the bijections that correspond to $b=-1$ were added in order to allow coupling moves in one direction (of a bijection $f^{(1)}_{i_1}$) on the $n_1$-bit long prefix with moves in the opposite direction (of a bijection $f^{(2)}_{i_2}$) on the $n_2$-bit long suffix.

Proof Sketch:

The expanding property of the combined 1-local graph, denoted G, can be seen by considering a sufficiently long random walk on it. The key observation is that a t-step random walk on G corresponds to two independent t-step random walks on the two 1-local graphs of the hypothesis, denoted $G_1$ and $G_2$. (In particular, a random step on G specifies a random tuple $(i_1,i_2,b)\in [d_1]\times [d_2]\times \{\pm 1\}$ and a direction $\delta \in \{\pm 1\}$ in which $f_{i_1,i_2,b}$ is applied, and this corresponds to random steps on the two graphs (i.e., a choice of $f^{(1)}_{i_1}$ and direction $\delta$ for $G_1$ and a choice of $f^{(2)}_{i_2}$ and direction $b\cdot \delta$ for $G_2$).) Hence, the $n_1$-bit long prefix (resp., the $n_2$-bit long suffix) of the end-vertex in a t-step random walk on G is $\exp (-\varOmega (t))$-close to the uniform distribution on $\{0,1\}^{n_1}$ (resp., on $\{0,1\}^{n_1}$), whereas these two parts are distributed independently of one another.

   $\blacksquare$