Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A (Weakly) Polynomial Algorithm for AIVF Coding

Reza Hosseini Dolatabadi, Mordecai J. Golin, Arian Zamani

TL;DR

The paper tackles designing maximum-cost AIVF codes for a fixed dictionary size by recasting the problem as a minimum-cost Markov chain (MCMC) problem and solving it with a linear programming approach based on the Ellipsoid method, establishing a (weakly) polynomial-time algorithm. It integrates dynamic programming for local optimization with a framework that reduces MCMC to LP via the construction in golinaifv-m, enabling a polynomial-time solution under standard bit-length bounds. The key contributions are (i) a rigorous mapping of AIVF coding to MCMC, (ii) a DP-based local optimization routine, and (iii) a theoretical polynomial-time algorithm for maximum-cost AIVF code construction, while noting that practical implementation remains challenging due to Ellipsoid-method inefficiencies. This work provides a foundational, theoretically efficient pathway for AIVF code design, with the open challenge of an efficient, practical polynomial-time algorithm.

Abstract

It is possible to improve upon Tunstall coding using a collection of multiple parse trees. The best such results so far are Iwata and Yamamoto's maximum cost AIVF codes. The most efficient algorithm for designing such codes is an iterative one that could run in exponential time. In this paper, we show that this problem fits into the framework of a newly developed technique that uses linear programming with the Ellipsoid method to solve the minimum cost Markov chain problem. This permits constructing maximum cost AIVF codes in (weakly) polynomial time.

A (Weakly) Polynomial Algorithm for AIVF Coding

TL;DR

The paper tackles designing maximum-cost AIVF codes for a fixed dictionary size by recasting the problem as a minimum-cost Markov chain (MCMC) problem and solving it with a linear programming approach based on the Ellipsoid method, establishing a (weakly) polynomial-time algorithm. It integrates dynamic programming for local optimization with a framework that reduces MCMC to LP via the construction in golinaifv-m, enabling a polynomial-time solution under standard bit-length bounds. The key contributions are (i) a rigorous mapping of AIVF coding to MCMC, (ii) a DP-based local optimization routine, and (iii) a theoretical polynomial-time algorithm for maximum-cost AIVF code construction, while noting that practical implementation remains challenging due to Ellipsoid-method inefficiencies. This work provides a foundational, theoretically efficient pathway for AIVF code design, with the open challenge of an efficient, practical polynomial-time algorithm.

Abstract

It is possible to improve upon Tunstall coding using a collection of multiple parse trees. The best such results so far are Iwata and Yamamoto's maximum cost AIVF codes. The most efficient algorithm for designing such codes is an iterative one that could run in exponential time. In this paper, we show that this problem fits into the framework of a newly developed technique that uses linear programming with the Ellipsoid method to solve the minimum cost Markov chain problem. This permits constructing maximum cost AIVF codes in (weakly) polynomial time.
Paper Structure (9 sections, 8 theorems, 29 equations, 3 figures, 2 algorithms)

This paper contains 9 sections, 8 theorems, 29 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Tunstall Coding
  3. AIVF Coding
  4. Dynamic Programming For AIVF
  5. The Minimum Cost Markov Chain Problem
  6. More Definitions
  7. The Main Result
  8. Dynamic Programming for AIVF:
  9. Proofs missing from the main paper

Key Result

Proposition 1

For any fixed $i$ and $\mathbf x$, dynamic programming can find $t_i\in\mathcal{T}_{i}^{(D)}$ maximizing ${\hbox{Cost}}(t_i,\mathbf x)$ in $O(D^2\cdot |S|)$ time.

Figures (3)

  • Figure 1: (Figure 1 in AIVF-iterative) The parse tree and dictionary $\mathcal{D}$ of the Tunstall code for source of $S=\{a_0,a_1,a_2\}$ with $p(a_0)=0.6$, $p(a_1)=0.3$, $p(a_2)=0.1$ and $D=7$.
  • Figure 2: (Figure 2 in AIVF-iterative) The optimal AIVF code for the source $S=\{a_0,a_1,a_2\}$ with $p(a_0)=0.6$, $p(a_1)=0.3$, $p(a_2)=0.1$ and $D=7$.
  • Figure 3: On the left-hand we can see an Illustration of how the trees in $\mathcal{T}_i$ can be decomposed for $i\in\mathcal{I}_{|S|-2}$ and on the right-hand side we can see how a the trees in $\mathcal{T}_{|S|-2}$ can be decomposed

Theorems & Definitions (17)

  • Proposition 1: AIVF-iterative
  • Definition 1
  • Proposition 2: Lemma 3.1 and Corollary 3.4 in golinaifv-m
  • Definition 2: $P$-restricted search spaces
  • Definition 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Remark 1
  • Theorem 1
  • ...and 7 more