Extracting Mergers and Projections of Partitions

TL;DR

This paper investigates the extraction of randomness from somewhere-random sources, introduces extracting mergers with constant seed length, and explores combinatorial partition problems related to Shearer's lemma.

Contribution

It establishes seed-length bounds for extracting mergers, shows the possibility of constant seed mergers for multiple bits, and introduces partition analogues of Shearer's lemma.

Findings

Seedless extracting mergers with one output bit do not exist.

Constant seed extracting mergers for multiple bits are possible, but random choices do not suffice.

Merging most entropy requires seed length of at least logarithmic in n.

Abstract

We study the problem of extracting randomness from somewhere-random sources, and related combinatorial phenomena: partition analogues of Shearer's lemma on projections. A somewhere-random source is a tuple $(X_1, \ldots, X_t)$ of (possibly correlated) $\{0,1\}^n$-valued random variables $X_i$ where for some unknown $i \in [t]$, $X_i$ is guaranteed to be uniformly distributed. An $extracting$ $merger$ is a seeded device that takes a somewhere-random source as input and outputs nearly uniform random bits. We study the seed-length needed for extracting mergers with constant $t$ and constant error. We show: $\cdot$ Just like in the case of standard extractors, seedless extracting mergers with even just one output bit do not exist. $\cdot$ Unlike the case of standard extractors, it $is$ possible to have extracting mergers that output a constant number of bits using only constant seed. Furthermore, a random choice of merger does not work for this purpose! $\cdot$ Nevertheless, just like in the case of standard extractors, an extracting merger which gets most of the entropy out (namely, having $\Omega$ $(n)$ output bits) must have $\Omega$ $(\log n)$ seed. This is the main technical result of our work, and is proved by a second-moment strengthening of the graph-theoretic approach of Radhakrishnan and Ta-Shma to extractors. In contrast, seed-length/output-length tradeoffs for condensing mergers (where the output is only required to have high min-entropy), can be fully explained by using standard condensers. Inspired by such considerations, we also formulate a new and basic class of problems in combinatorics: partition analogues of Shearer's lemma. We show basic results in this direction; in particular, we prove that in any partition of the $3$-dimensional cube $[0,1]^3$ into two parts, one of the parts has an axis parallel $2$-dimensional projection of area at least $3/4$.