Worst-Case Optimal Adaptive Prefix Coding
Abstract
A common complaint about adaptive prefix coding is that it is much slower than static prefix coding. Karpinski and Nekrich recently took an important step towards resolving this: they gave an adaptive Shannon coding algorithm that encodes each character in (O (1)) amortized time and decodes it in (O ( H)) amortized time, where H is the empirical entropy of the input string s. For comparison, Gagie's adaptive Shannon coder and both Knuth's and Vitter's adaptive Huffman coders all use ( (H)) amortized time for each character. In this paper we give an adaptive Shannon coder that both encodes and decodes each character in (O (1)) worst-case time. As with both previous adaptive Shannon coders, we store s in at most ((H + 1) |s| + o (|s|)) bits. We also show that this encoding length is worst-case optimal up to the lower order term.