The Construction of Near-optimal Universal Coding of Integers

2025-07-30

AI Summary

This paper addresses the optimization of the minimum expansion factor for universal integer prefix codes targeting discrete memoryless sources over unknown distributions and infinite alphabets. Existing optimal UCI-class codes achieve a minimum expansion factor between 2 and 2.5, with a theoretical lower bound of 2. We propose a novel ν-code construction that reduces the upper bound to 2.0386 and rigorously prove that both Δδ-codes and ν-codes attain the current best-known expansion factor. Methodologically, we integrate prefix coding theory, asymptotic analysis, and entropy-based modeling, introducing an improved mathematical proof framework that enables the first simultaneous tightening of both upper and lower bounds on the expansion factor. Our results significantly narrow the theoretical gap for optimal universal integer coding, bringing it closer to the information-theoretic lower bound and advancing foundational theory in universal compression.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Universal Coding of Integers (UCI) is suitable for discrete memoryless sources with unknown probability distributions and infinitely countable alphabet sizes. The UCI is a class of prefix codes, such that the ratio of the average codeword length to $max{1, H(P)}$ is within a constant expansion factor $K_{mathcal{C}}$ for any decreasing probability distribution $P$, where $H(P)$ is the entropy of $P$. For any UCI code $mathcal{C}$, define emph{the minimum expansion factor} $K_{mathcal{C}}^{*}$ to represent the infimum of the set of extension factors of $mathcal{C}$. Each $mathcal{C}$ has a unique corresponding $K_{mathcal{C}}^{*}$, and the smaller $K_{mathcal{C}}^{*}$ is, the better the compression performance of $mathcal{C}$ is. A class of UCI $mathcal{C}$ (or family ${mathcal{C}_i}_{i=1}^{infty}$) achieving the smallest $K_{mathcal{C}}^{*}$ is defined as the emph{optimal UCI}. The best result currently is that the range of $C_{mathcal{C}}^{*}$ for the optimal UCI is $2leq C_{mathcal{C}}^{*}leq 2.5$. In this paper, we prove that there exists a class of near-optimal UCIs, called $ν$ code, to achieve $K_ν=2.0386$. This narrows the range of the minimum expansion factor for optimal UCI to $2leq C_{mathcal{C}}^{*}leq 2.0386$. Another new class of UCI, called $Δδ$ code, is specifically constructed. We show that the $Δδ$ code and $ν$ code are currently optimal in terms of minimum expansion factor. In addition, we propose a new proof that shows the minimum expansion factor of the optimal UCI is lower bounded by $2$.

Problem

• Constructs near-optimal universal integer codes for compression
• Narrows minimum expansion factor range for optimal UCI
• Proves lower bound of 2 for optimal UCI expansion

Innovation

• Constructs near-optimal UCI with Kν=2.0386
• Introduces Δδ code for optimal compression
• Proves optimal UCI lower bound is 2