Fast and Compact Prefix Codes

It is well-known that, given a probability distribution over $n$ characters, in the worst case it takes (\Theta (n \log n)) bits to store a prefix code with minimum expected codeword length. However, in this paper we first show that, for any $0<ϵ<1\mathrm{/}2$ with (1 / \epsilon = \Oh{\polylog{n}}), it takes $\text{Oh}n\log \log (1\mathrm{/}ϵ)$ bits to store a prefix code with expected codeword length within $ϵ$ of the minimum. We then show that, for any constant (c > 1), it takes $\text{Oh}{n}^{1\mathrm{/}c}\log n$ bits to store a prefix code with expected codeword length at most $c$ times the minimum. In both cases, our data structures allow us to encode and decode any character in $\text{Oh}1$ time.