$O(\log \log n)$ Worst-Case Local Decoding and Update Efficiency for Data Compression

Abstract

This paper addresses the problem of data compression with local decoding and local update. A compression scheme has worst-case local decoding $ d_{wc} $ if we can recover any bit of the raw file by probing at most $ d_{wc} $ bits of the compressed sequence, and has update efficiency of $u_{wc} $ if a single bit of the raw file can be updated by modifying at most $ u_{wc} $ bits of the compressed sequence. This article provides an entropy-achieving compression scheme for memoryless sources that simultaneously achieves $ O(\log\log n) $ local decoding and update efficiency. Key to this achievability result is a novel succinct data structure for sparse sequences which allows efficient local decoding and local update. Under general assumptions on the local decode and update algorithms, a converse result shows that the maximum of $ d_{wc} $ and $ u_{wc} $ must grow as $ \Omega(\log\log n) $.