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Linear Network Coding for Robust Function Computation and Its Applications in Distributed Computing

Hengjia Wei, Min Xu, Gennian Ge

TL;DR

This paper presents a minimum distance decoder for linear network codes and focuses on the sum function and the identity function, showing that in any directed acyclic network there are two classes of linear network codes for these target functions, respectively, that attain the Singleton-like bound.

Abstract

We investigate linear network coding in the context of robust function computation, where a sink node is tasked with computing a target function of messages generated at multiple source nodes. In a previous work, a new distance measure was introduced to evaluate the error tolerance of a linear network code for function computation, along with a Singleton-like bound for this distance. In this paper, we first present a minimum distance decoder for these linear network codes. We then focus on the sum function and the identity function, showing that in any directed acyclic network there are two classes of linear network codes for these target functions, respectively, that attain the Singleton-like bound. Additionally, we explore the application of these codes in distributed computing and design a distributed gradient coding scheme in a heterogeneous setting, optimizing the trade-off between straggler tolerance, computation cost, and communication cost. This scheme can also defend against Byzantine attacks.

Linear Network Coding for Robust Function Computation and Its Applications in Distributed Computing

TL;DR

This paper presents a minimum distance decoder for linear network codes and focuses on the sum function and the identity function, showing that in any directed acyclic network there are two classes of linear network codes for these target functions, respectively, that attain the Singleton-like bound.

Abstract

We investigate linear network coding in the context of robust function computation, where a sink node is tasked with computing a target function of messages generated at multiple source nodes. In a previous work, a new distance measure was introduced to evaluate the error tolerance of a linear network code for function computation, along with a Singleton-like bound for this distance. In this paper, we first present a minimum distance decoder for these linear network codes. We then focus on the sum function and the identity function, showing that in any directed acyclic network there are two classes of linear network codes for these target functions, respectively, that attain the Singleton-like bound. Additionally, we explore the application of these codes in distributed computing and design a distributed gradient coding scheme in a heterogeneous setting, optimizing the trade-off between straggler tolerance, computation cost, and communication cost. This scheme can also defend against Byzantine attacks.
Paper Structure (9 sections, 19 theorems, 103 equations, 3 figures)

This paper contains 9 sections, 19 theorems, 103 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Network function computation model
  4. Robust network function computation model
  5. Decoding for Robust Network Function computation
  6. Computing the Sum Function Against Errors
  7. Computing the Identity Function Against Errors
  8. Bounds on Robust Computing Capacity
  9. Applications in Distributed Computing

Key Result

Lemma 2.1

Given a network $\mathcal{N}$ and a linear target function $f(\mathbf{x})=\mathbf{x} \cdot T$ with $T\in \mathbb{F}_q^{s\times l}$. If there exists a $(k,n)$ network code $\mathcal{C}$ computing $f$ with rate $k/n$, then necessarily where $T_{I_C}$ is the $\lvert I_C\rvert\times l$ submatrix of $T$ which is obtained by choosing the rows of $T$ indexed by $I_C$.

Figures (3)

  • Figure 1: The network on the left is a sum-network, where each source $\sigma_i$ generates a message $x_i$ and the sink node wants to compute the sum $x_1+x_2$. The network on the right is a multicast network, along with a coding scheme which achieves the maximum communication rate $3$.
  • Figure 2: A coding scheme designed to compute $x_1+x_2$ over $\mathbb{F}_q$, where $q$ is odd.
  • Figure 3: A coding scheme designed to compute $x_1+x_2$ over $\mathbb{F}_q$, where $q$ is even.

Theorems & Definitions (21)

  • Lemma 2.1: WeiXuGe23
  • Theorem 2.1: WeiXuGe23
  • Theorem 2.2: WeiXuGe23
  • Theorem 2.3: WeiXuGe23
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.1
  • Example 4.1
  • Lemma 4.1
  • Theorem 4.1
  • ...and 11 more