Compressing Sparse Sequences under Local Decodability Constraints

TL;DR

This paper studies how to efficiently encode sparse binary sequences with local decodability constraints, providing bounds on blocklength scaling and exploring adaptive and non-adaptive models, filling a gap in fixed-blocklength understanding.

Contribution

It derives upper and lower bounds on blocklength scaling for locally decodable codes of sparse sequences, addressing an open problem in fixed-blocklength scenarios.

Findings

Bounds often coincide up to constant factors

Characterization of scaling behavior under local decodability constraints

Connections to communication complexity are discussed

Abstract

We consider a variable-length source coding problem subject to local decodability constraints. In particular, we investigate the blocklength scaling behavior attainable by encodings of $r$-sparse binary sequences, under the constraint that any source bit can be correctly decoded upon probing at most $d$ codeword bits. We consider both adaptive and non-adaptive access models, and derive upper and lower bounds that often coincide up to constant factors. Notably, such a characterization for the fixed-blocklength analog of our problem remains unknown, despite considerable research over the last three decades. Connections to communication complexity are also briefly discussed.