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Distributed matrix multiplication with straggler tolerance using algebraic function fields

Adrián Fidalgo-Díaz, Umberto Martínez-Peñas

TL;DR

This work advances straggler mitigation in distributed matrix multiplication by replacing classical polynomial-based codes with algebraic-geometry codes defined over algebraic function fields. By leveraging the Weierstrass semigroup and a carefully chosen AG basis, it develops AG polynomial codes and AG matdot codes that support a larger number of workers over small fields while sustaining comparable per-worker computation. The authors provide explicit construction strategies (via Apéry sets and semigroup structure) and establish both optimal and near-optimal recovery thresholds; they also analyze decoding complexity and demonstrate asymptotically near-optimal performance using García–Stichtenoth towers. The results offer scalable, efficient DMM schemes with robust straggler tolerance and practical applicability for large-scale distributed systems.

Abstract

The problem of straggler mitigation in distributed matrix multiplication (DMM) is considered for a large number of worker nodes and a fixed small finite field. Polynomial codes and matdot codes are generalized by making use of algebraic function fields (i.e., algebraic functions over an algebraic curve) over a finite field. The construction of optimal solutions is translated to a combinatorial problem on the Weierstrass semigroups of the corresponding algebraic curves. Optimal or almost optimal solutions are provided. These have the same computational complexity per worker as classical polynomial and matdot codes, and their recovery thresholds are almost optimal in the asymptotic regime (growing number of workers and a fixed finite field).

Distributed matrix multiplication with straggler tolerance using algebraic function fields

TL;DR

This work advances straggler mitigation in distributed matrix multiplication by replacing classical polynomial-based codes with algebraic-geometry codes defined over algebraic function fields. By leveraging the Weierstrass semigroup and a carefully chosen AG basis, it develops AG polynomial codes and AG matdot codes that support a larger number of workers over small fields while sustaining comparable per-worker computation. The authors provide explicit construction strategies (via Apéry sets and semigroup structure) and establish both optimal and near-optimal recovery thresholds; they also analyze decoding complexity and demonstrate asymptotically near-optimal performance using García–Stichtenoth towers. The results offer scalable, efficient DMM schemes with robust straggler tolerance and practical applicability for large-scale distributed systems.

Abstract

The problem of straggler mitigation in distributed matrix multiplication (DMM) is considered for a large number of worker nodes and a fixed small finite field. Polynomial codes and matdot codes are generalized by making use of algebraic function fields (i.e., algebraic functions over an algebraic curve) over a finite field. The construction of optimal solutions is translated to a combinatorial problem on the Weierstrass semigroups of the corresponding algebraic curves. Optimal or almost optimal solutions are provided. These have the same computational complexity per worker as classical polynomial and matdot codes, and their recovery thresholds are almost optimal in the asymptotic regime (growing number of workers and a fixed finite field).
Paper Structure (10 sections, 13 theorems, 81 equations, 2 figures, 2 tables)

This paper contains 10 sections, 13 theorems, 81 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algebraic-geometry polynomial codes
  4. The classical solution
  5. The Algebraic-Geometry solution
  6. Algebraic-geometry matdot codes
  7. The classical solution
  8. The Algebraic-Geometry solution
  9. Decoding and complexity
  10. Asymptotic analysis of the recovery threshold

Key Result

Lemma 1

Let $S$ be a numerical semigroup and $n \in S^*$. Then

Figures (2)

  • Figure 1: General scheme of DMM with straggler tolerance, where each $\tilde{A}_i$ and $\tilde{B}_i$ denotes a matrix of lower size than $A$ and $B$.
  • Figure 2: Commutative diagram summarizing the polynomial code algorithm.

Theorems & Definitions (46)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Remark 1
  • Remark 2
  • Definition 7
  • Lemma 1
  • ...and 36 more