Oblivious Tight Compaction In O(n) Time with Smaller Constant

Abstract

Oblivious compaction is a crucial building block for hash-based oblivious RAM. Asharov et al. recently gave a O(n) algorithm for oblivious tight compaction [2]. Their algorithm is deterministic and asymptotically optimal, but the implied constant is ${\gg }2^{111}$. We give a new algorithm for oblivious tight compaction that runs in time ${<}23913.17n + o(n)$. As part of our construction, we give a new result in the bootstrap percolation of random regular graphs.

Appendices

A Margulis Graph Compaction

1.1 A.1 Analysis of Algorithm 11

The argument here is similar to the argument in § 5.1, and is given in full in the full version of this paper [12].

B Combinatorial Lemmas

We state here a number of combinatorial lemmas we need throughout the paper, and give a citation if the result is well-known. For proofs and discussion, we refer to the full version of this paper [12].

Lemma 1

[Margulis [22]] There is a family of 4-regular expander graphs $\{G_k\}_{k \ge 0}$ with girth satisfying $|G_k| \ge 0.756 \log G_k$, with $G_k = (p_k^3-p_k)$, for $p_k$ the kth prime.

The following result is due to Friedman [13] proving a conjecture of Alon.

Lemma 2

(Friedman). The probability that a random d-regular graph G on n vertices, with dn even and $n > d$, has second eigenvalue ${\le }2\sqrt{d-1}+\epsilon$, is $1-O(n^{-c})$.

The following is due to Steger and Wormald [32], though the algorithm was first studied by Rucinski and Wormald [29].

Lemma 3

(Steger, Wormald). There is an algorithm for generating a random d-regular graph, uniformly, in time $O(nd^2)$.

Lemma 4

It is possible to test whether a d-regular graph G on n vertices is Ramanujan in time poly(n).

Combining these three results gives us an algorithm for generating a d-regular expander graph on n vertices in time poly(n). Since our expander graphs will have $O(\log n)$ vertices, and we only need to generate  $O(\log n)$ such graphs, this is sufficient for our needs.

Lemma 5

Consider a random walk along the integers $\mathbb {Z}$ where the probability of moving from i to $i+1$ is $f(\frac{i}{n})$ for some increasing function f. Let X be a random variable denoting the time it takes to walk from $\rho n$ to $\rho ' n$. Then

$$\Pr \left( \big |X - n\int _{\rho }^{\rho '} \frac{dx}{f(x)}\big | > \epsilon n^{3/4} \right) = e^{-8(\rho '-\rho ) \epsilon ^2\sqrt{n}} .$$

Lemma 6

Fix constants $\gamma ' < \gamma$, and let A be an array of length n with $\gamma ' n$ marked balls. Then if $k\log n / \log \log n$ balls are chosen at random from A, the probability there are more than $\gamma k\log n$ marked balls chosen is $n^{-\frac{(\gamma -\gamma ')^2k}{(\gamma +\gamma ') \log \log n}}$.

Lemma 7

Fix a constant $c < 1$, and let A be an array of $\varOmega (n/\log ^3 n)$ balls with at most $|A|\exp \left( (c-1) \log n (\log \log n)^t\right)$ marked balls, for $t > -2$. Then if $k\log n/\log \log n$ balls are chosen at random from A, the probability there are more than $\ell \log \log n$ marked balls chosen is $\exp \left( -O(\log n (\log \log n)^{t+1})\right)$, for every choice of $k,\ell > 0$. In particular, for $t = 0$, it is negligible in n.

Lemma 8

Fix constants $\varepsilon$, $\gamma$ and k. Let A be an array of kn balls, with $\gamma kn$ balls marked. Shuffle A and divide it into bins of size k. The probability that there are fewer than $\left( (1-\gamma )^k-\varepsilon \right) n$ empty bins is $e^{-cn}$, for some constant c.

C Proof of Theorem 2

Proof

Choose $c > \frac{1}{2}$ and suppose by way of contradiction there is some set U with $\frac{|U|}{|G|} > c$ that does not activate all of G under percolation. Let S be the set that is activated by U. Then we also have $\frac{|S|}{|G|} > c$, and all vertices in $T = G \backslash S$ have fewer than $\lceil \frac{d+1}{2} \rceil$ edges to S. By the expander mixing lemma (see e.g. [17]) we have

$$\begin{aligned} e(T,T)&\le \frac{d|T|^2}{|G|} + \lambda |T| \\&\le \left( \frac{d|T|}{|G|}+2\sqrt{d-1}\right) |T|, \end{aligned}$$

when G is an expander graph. Taking $T = G \backslash S$ gives

$$\begin{aligned} e(T,T)&\le \left( d-\frac{dS}{G}+2\sqrt{d-1}\right) |T|\\&\le \left( d(1-c)+2\sqrt{d-1}\right) |T|. \end{aligned}$$

First consider the case when d is odd. We have

$$ 2\sqrt{d-1} < \frac{d+1}{2} $$

for $d \ge 13$, so that, for c sufficiently close to 1, we have

$$ e(T,T) < \frac{d+1}{2}|T| $$

and

$$ e(S,T) > \frac{d-1}{2}|T|. $$

Thus at least one vertex of T has at least $\frac{d+1}{2}$ edges to S, and can be activated, a contradiction.

The proof for d even is similar, and relies on the fact that $d=16$ is the smallest value with

$$ 2\sqrt{d-1} < \frac{d}{2}. $$

Corollary C1

For $d=13$, there is some constant $\gamma$ such that every set S with $|S|/|G| \ge (1-\gamma )^{d+1}$ is a percolating set for majority bootstrap percolation.

Proof

This is an immediate consequence of Theorem 2 but we derive $\gamma$ explicitly. We desire

$$ \left( d-d(1-\gamma )^{d+1}+2\sqrt{d-1}\right) < \frac{d+1}{2}, $$

and so choose $\gamma < 0.005307$.