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Sparse Degree Optimization for BATS Codes

Hoover H. F. Yin, Jie Wang

TL;DR

This paper investigates the sparsity optimization of degree distribution for BATS codes that produces sparse degree distributions and discusses a few heuristics and also a way to obtain an exact sparsity solution.

Abstract

Batched sparse (BATS) code is a class of batched network code that can achieve a close-to-optimal rate when an optimal degree distribution is provided. We observed that most probability masses in this optimal distribution are very small, i.e., the distribution "looks" sparse. In this paper, we investigate the sparsity optimization of degree distribution for BATS codes that produces sparse degree distributions. There are many advantages to use a sparse degree distribution, say, it is robust to precision errors when sampling the degree distribution during encoding and decoding in practice. We discuss a few heuristics and also a way to obtain an exact sparsity solution. These approaches give a trade-off between computational time and achievable rate, thus give us the flexibility to adopt BATS codes in various scenarios, e.g., device with limited computational power, stable channel condition, etc.

Sparse Degree Optimization for BATS Codes

TL;DR

This paper investigates the sparsity optimization of degree distribution for BATS codes that produces sparse degree distributions and discusses a few heuristics and also a way to obtain an exact sparsity solution.

Abstract

Batched sparse (BATS) code is a class of batched network code that can achieve a close-to-optimal rate when an optimal degree distribution is provided. We observed that most probability masses in this optimal distribution are very small, i.e., the distribution "looks" sparse. In this paper, we investigate the sparsity optimization of degree distribution for BATS codes that produces sparse degree distributions. There are many advantages to use a sparse degree distribution, say, it is robust to precision errors when sampling the degree distribution during encoding and decoding in practice. We discuss a few heuristics and also a way to obtain an exact sparsity solution. These approaches give a trade-off between computational time and achievable rate, thus give us the flexibility to adopt BATS codes in various scenarios, e.g., device with limited computational power, stable channel condition, etc.
Paper Structure (14 sections, 2 theorems, 16 equations, 5 figures, 1 table, 2 algorithms)

This paper contains 14 sections, 2 theorems, 16 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. BATS Codes
  3. Encoding
  4. Recoding and Decoding
  5. Degree Distribution Optimizations
  6. Sparsity Optimizations
  7. Via Direct Trimming
  8. Via Complementary Slackness
  9. Iterative Reweighted $\ell_1$-Norm Heuristic
  10. Exact Solver
  11. Numerical Evaluations
  12. Concluding Remarks
  13. Proofs of Technical Results
  14. Additional Numerical Study

Key Result

Proposition 1

Assume that as $m\to\infty$, $\epsilon_m\downarrow 0$ and $\rho_m\downarrow0$. Then any accumulation point of the sequence $\{\theta_m\}_m$ is an optimal solution to Eq:optimization:nominal.

Figures (5)

  • Figure 1: Illustrations of optimal degree distributions.
  • Figure 2: Comparison of optimal value (in left $y$-axis) and computational time (in right $y$-axis) with various choices of $|\mathcal{X}|$ for exact degree optimization solver. The naive Gurobi solver successfully solves only the first $5$ instances within $10^4$ seconds, whereas our customized algorithm is quite scalable for large problem size.
  • Figure 3: Comparison of optimal value (in left $y$-axis) and computational time (in right $y$-axis) with various choices of sparsity level $s$ for exact degree optimization solver.
  • Figure : Iterative Reweighted $\ell_1$-Norm Heuristic
  • Figure : Optimal Sparsity Vector via Problem \ref{['Eq:max:fz']}

Theorems & Definitions (2)

  • Proposition 1: Convergence of Discretization
  • Theorem 1: Convergence Rate of Discretization