The Effect of Local Decodability Constraints on Variable-Length Compression

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 nonadaptive access models, and derive upper and lower bounds that often coincide up to constant factors. Such a characterization for the fixed-blocklength analog of our problem, known as the bit probe complexity of static membership, remains unknown despite considerable attention from researchers over the last few decades. We also show that locally decodable schemes for sparse sequences are able to decode 0s (frequent source symbols) of the source with far fewer probes on average than they can decode 1s (infrequent source symbols), thus rigorizing the notion that infrequent symbols require high probe complexity, even on average. Connections to the fixed-blocklength model and to communication complexity are also briefly discussed.