[Paper Review] The Optimality of AIFV Codes in the Class of $2$-bit Delay Decodable Codes

Kengo Hashimoto, Iwata Ken-ichi|arXiv (Cornell University)|Jun 16, 2023

DNA and Biological ComputingBiochemistry, Genetics and Molecular Biology3 citations

TL;DR

This paper proves that AIFV (Almost Instantaneous Fixed-to-Variable) codes achieve the optimal average codeword length among all 2-bit delay decodable codes, regardless of the number of code tables used. By generalizing prior results that were limited to two-code-table systems, the authors establish AIFV codes as the optimal choice in the broader class of finite-code-table, 2-bit delay decodable source codes, using a formal framework of code-tuples and delay constraints.

ABSTRACT

AIFV (almost instantaneous fixed-to-variable length) codes are noiseless source codes that can attain a shorter average codeword length than Huffman codes by allowing a time-variant encoder with two code tables and a decoding delay of at most 2 bits. First, we consider a general class of noiseless source codes, called k-bit delay decodable codes, in which one allows a finite number of code tables and a decoding delay of at most k bits for k >= 0. Then we prove that AIFV codes achieve the optimal average codeword length in the 2-bit delay decodable codes class.

Motivation & Objective

• To establish the optimality of AIFV codes in the general class of 2-bit delay decodable codes with any finite number of code tables.
• To resolve the open problem of whether AIFV codes remain optimal when more than two code tables are allowed under a 2-bit decoding delay constraint.
• To formalize the class of k-bit delay decodable code-tuples and analyze their structural and optimality properties.
• To generalize prior results limited to two-code-table systems to the full class of finite-code-table, 2-bit delay decodable codes.

Proposed method

• The authors define a code-tuple as a system using a finite number of code tables and a state transition mechanism governed by mappings τi.
• They introduce the concept of k-bit delay decodable code-tuples, where decoding of a symbol depends on at most k bits of future information.
• The paper uses a formal framework based on prefix sets and transition probabilities to model the behavior of code-tuples under delay constraints.
• It defines key components such as P^k_{F,i}, the set of k-bit prefixes that can lead to a symbol being decoded with delay k.
• The proof relies on structural analysis of regular and extendable code-tuples, and introduces auxiliary code-tuples like F̂, Ḟ, and F̈ to simplify the optimality argument.
• The main result is derived through a series of lemmas and case analysis, culminating in the proof that AIFV codes are optimal in the 2-bit delay decodable class.

Experimental results

Research questions

• RQ1 Is the AIFV code optimal among all 2-bit delay decodable codes when more than two code tables are allowed?
• RQ2 Can the optimality of AIFV codes, previously shown only for two-code-table systems, be extended to the general class of finite-code-table, 2-bit delay decodable codes?
• RQ3 What structural properties must a code-tuple satisfy to be optimal under a 2-bit delay constraint?
• RQ4 How do the transition probabilities and prefix sets P^k_{F,i} influence the average codeword length in delay-constrained coding systems?

Key findings

• AIFV codes achieve the minimum possible average codeword length among all 2-bit delay decodable codes, regardless of the number of code tables used.
• The optimality of AIFV codes holds even when the number of code tables exceeds two, resolving a previously open question.
• The proof establishes that AIFV codes are optimal in the class of regular, extendable, and 2-bit delay decodable code-tuples.
• The authors show that any code-tuple with a 2-bit delay constraint can be transformed into an equivalent AIFV code without increasing the average codeword length.
• The result confirms that AIFV codes are not only optimal for two-table systems but are the optimal choice in the entire 2-bit delay decodable class.
• The paper provides a complete characterization of the optimality conditions for 2-bit delay decodable codes, with AIFV codes forming the optimal subclass.