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Better Algorithms for Constructing Minimum Cost Markov Chains and AIFV Codes

Reza Hosseini Dolatabadi, Mordedcai J. Golin, Arian Zamani

TL;DR

This work addresses constructing minimum-cost Markov Chains with applications to $AIFV$ coding, reframing the optimization as a Markov Chain Polytope problem. It provides a complete termination and correctness proof for the existing iterative algorithm, strengthening theoretical guarantees. It also presents a new, simpler weakly polynomial-time approach for $AIFV$-$m$ coding by exploiting MCP structure and replacing the Ellipsoid method with binary search, achieving practical efficiency. For the special case of $AIFV$-$3$ coding, the authors derive a concrete $O(n^5 b^2)$ time algorithm, with potential extension to larger $m$, showcasing substantial improvements in both theory and practice for near-optimal, rapidly computable codes.

Abstract

The problem of constructing optimal AIFV codes is a special case of that of constructing minimum cost Markov Chains. This paper provides the first complete proof of correctness for the previously known iterative algorithm for constructing such Markov chains. A recent work describes how to efficiently solve the Markov Chain problem by first constructing a Markov Chain Polytope and then running the Ellipsoid algorithm for linear programming on it. This paper's second result is that, in the AIFV case, a special property of the polytope instead permits solving the corresponding linear program using simple binary search

Better Algorithms for Constructing Minimum Cost Markov Chains and AIFV Codes

TL;DR

This work addresses constructing minimum-cost Markov Chains with applications to coding, reframing the optimization as a Markov Chain Polytope problem. It provides a complete termination and correctness proof for the existing iterative algorithm, strengthening theoretical guarantees. It also presents a new, simpler weakly polynomial-time approach for - coding by exploiting MCP structure and replacing the Ellipsoid method with binary search, achieving practical efficiency. For the special case of - coding, the authors derive a concrete time algorithm, with potential extension to larger , showcasing substantial improvements in both theory and practice for near-optimal, rapidly computable codes.

Abstract

The problem of constructing optimal AIFV codes is a special case of that of constructing minimum cost Markov Chains. This paper provides the first complete proof of correctness for the previously known iterative algorithm for constructing such Markov chains. A recent work describes how to efficiently solve the Markov Chain problem by first constructing a Markov Chain Polytope and then running the Ellipsoid algorithm for linear programming on it. This paper's second result is that, in the AIFV case, a special property of the polytope instead permits solving the corresponding linear program using simple binary search
Paper Structure (6 sections, 14 theorems, 47 equations, 5 figures, 2 algorithms)

This paper contains 6 sections, 14 theorems, 47 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Minimum Cost Markov Chain Problem
  3. The Markov Chain Polytope
  4. The Iterative Algorithm
  5. Proof of Correctness of Algorithm \ref{['alg: iterative']}
  6. A New Weakly Polynomial Time Algorithm For AIFV-$3$-coding

Key Result

Proposition 1

Let $\mathbf S =(S_0,\ldots,S_{m-1})$ be any permissible Markov chain and $f_k \left( \mathbf x,S_i \right)$, $k \in [m],$ its associated hyperplanes. Then these $m$ hyperplanes intersect at a unique point $(\mathbf x,y) \in \mathbb{R}^{m}$. Furthermore, $y \ge {\rm height}(\mathbb{H}).$ Such a poin

Figures (5)

  • Figure 1: Node types in a binary AIFV-$3$ code tree: complete node ($C$), intermediate-$0$ and intermediate-$1$ nodes ($I_0$, $I_1$), master nodes of degrees $0, 1, 2$ ($M_0$, $M_1$, $M_2$), ($M_0$ is a leaf)and non-intermediate-$0$ nodes ($\neg{I_0}$). The $\neg {I_0}$ nodes can be complete, master, or intermediate-$1$ nodes, depending upon their location in the tree.
  • Figure 2: Example binary AIFV-$3$ code for source alphabet $\left\{ a, b, c, d\right\}.$ The small nodes are either complete or intermediate nodes, while the large nodes are master nodes with their assigned source symbols. Note that $T_2$ encodes $\hat{a}$, which is at its root, with an empty string!
  • Figure 3: Encoding $c b a b$ using the binary AIFV-$3$ code in Figure \ref{['fig:example_aifv_3X']}. The first $c$ is encoded using a degree-$0$ master node (leaf) in $T_0$ so the first $b$ is encoded using $T_0.$ This $b$ is encoded using a degree-$2$ master node, so $a$ is encoded using $T_2.$$a$ is encoded using a degree-$1$ master node, so the second $b$ is encoded using $T_1\ldots.$
  • Figure 4: Decoding $0001010$ using the binary AIFV-$3$ code in Figure \ref{['fig:example_aifv_3X']}.
  • Figure 5: Markov chain corresponding to AIFV-$3$ code in Figure \ref{['fig:example_aifv_3X']}. Note that $T_1$ contains no degree 1 master node, so there is no edge from $T_1$ to $T_1.$ Similarly, $T_2$ contains no degree 2 master node, so there is no edge from $T_2$ to $T_2.$

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Proposition 1: Lemma 3.1 in golinaifv-m
  • Corollary 1
  • Definition 3
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 22 more