Efficient and Compact Representations of Some Non-Canonical Prefix-Free Codes

TL;DR

This paper introduces efficient, compact data structures for storing and processing various types of prefix-free codes, enabling fast encoding and decoding with minimal space, applicable in the word-RAM computational model.

Contribution

It presents novel space-efficient representations for optimal and nearly optimal prefix-free codes, including alphabetic and lexicographically ordered variants, with improved encoding and decoding times.

Findings

Optimal alphabetic prefix-free code stored in O(σ log L) bits

Encoding and decoding in O(min(ℓ, log L)) time

Nearly optimal alphabetic code stored in o(σ) bits with constant-time operations

Abstract

For many kinds of prefix-free codes there are efficient and compact alternatives to the traditional tree-based representation. Since these put the codes into canonical form, however, they can only be used when we can choose the order in which codewords are assigned to symbols. In this paper we first show how, given a probability distribution over an alphabet of $\sigma$ symbols, we can store an optimal alphabetic prefix-free code in $\Oh{\sigma \log L}$ bits such that we can encode and decode any codeword of length $\ell$ in $\Oh{\min (\ell, \log L)}$ time, where $L$ is the maximum codeword length. With $\Oh{2^{L^\epsilon}}$ further bits, for any constant $\epsilon>0$, we can encode and decode $\Oh{\log \ell}$ time. We then show how to store a nearly optimal alphabetic prefix-free code in \(o (\sigma)\) bits such that we can encode and decode in constant time. We also consider a kind of optimal prefix-free code introduced recently where the codewords' lengths are non-decreasing if arranged in lexicographic order of their reverses. We reduce their storage space to $\Oh{\sigma \log L}$ while maintaining encoding and decoding times in $\Oh{\ell}$. We also show how, with $\Oh{2^{\epsilon L}}$ further bits, we can encode and decode in constant time. All of our results hold in the word-RAM model.