Worst-case Redundancy of Optimal Binary AIFV Codes and their Extended Codes

TL;DR

This paper improves the upper bound on the worst-case redundancy of optimal binary AIFV codes from 1 to 1/2, and introduces extended codes with reduced redundancy up to 1/m for m ≤ 4.

Contribution

It establishes a tighter bound on the worst-case redundancy of binary AIFV codes and proposes extended codes with lower redundancy and controlled decoding delay.

Findings

Worst-case redundancy bound improved to 1/2

Extended codes achieve redundancy of 1/m for m ≤ 4

Redundancy depends on the maximum source symbol probability

Abstract

Binary AIFV codes are lossless codes that generalize the class of instantaneous FV codes. The code uses two code trees and assigns source symbols to incomplete internal nodes as well as to leaves. AIFV codes are empirically shown to attain better compression ratio than Huffman codes. Nevertheless, an upper bound on the redundancy of optimal binary AIFV codes is only known to be 1, which is the same as the bound of Huffman codes. In this paper, the upper bound is improved to 1/2, which is shown to coincide with the worst-case redundancy of the codes. Along with this, the worst-case redundancy is derived in terms of $p_{\max}\geq$1/2, where $p_{\max}$ is the probability of the most likely source symbol. Additionally, we propose an extension of binary AIFV codes, which use $m$ code trees and allow at most $m$-bit decoding delay. We show that the worst-case redundancy of the extended binary AIFV codes is $1/m$ for $m \leq 4.$