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arxiv: 1906.10105 · v1 · pith:YM77A5P3new · submitted 2019-06-24 · 💻 cs.IT · cs.DC· math.IT

Coded Distributed Computing: Performance Limits and Code Designs

Mohammad Vahid Jamali , Mahdi Soleymani , Hessam Mahdavifar This is my paper

Pith reviewed 2026-05-25 16:45 UTC · model grok-4.3

classification 💻 cs.IT cs.DCmath.IT
keywords coded distributed computinglinear codeserasure channelsaverage execution timeperformance limitspolar codesReed-Muller codesmatrix multiplication
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The pith

Coded distributed computing execution time equals the error probability of linear codes over erasure channels.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper links the average time to complete a large linear job split across n nodes to the probability that an (n,k) linear code fails to recover the result over an erasure channel. This equivalence produces closed-form expressions for the execution time achieved by random binary linear codes and for the best time attainable by any linear code. It also proves that certain binary linear codes reach the optimal linear-code performance in the large-n limit. The approach imports results from coding theory to design faster distributed systems for tasks such as matrix multiplication.

Core claim

The average execution time of a coded distributed computing system that encodes k tasks with an (n,k) linear code is identical to the codeword error probability of that code transmitted over an erasure channel; this identity yields exact expressions for random binary linear codes, identifies the minimal time any linear code can achieve, and shows that good binary linear codes attain the same asymptotic optimum as the best linear code over any alphabet.

What carries the argument

Equivalence between average system completion time and codeword error probability of an (n,k) linear code over an erasure channel.

If this is right

  • Closed-form expressions exist for the average execution time realized by binary random linear codes.
  • The minimal average execution time achievable by any linear code is characterized exactly.
  • Binary linear codes exist that match, asymptotically, the best performance of any linear code.
  • Polar and Reed-Muller codes can be substituted to trade decoding complexity against performance.
  • Any advance in erasure-channel code design immediately supplies a candidate for faster distributed computing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Designers can now optimize node count n and code rate by consulting tables of known erasure-code performance rather than running Monte-Carlo simulations of the computing system.
  • The same modeling step could be tested on non-linear computations if their failure modes can be mapped to an erasure pattern.
  • Low-complexity decoders already developed for polar codes become directly usable for recovering partial results in distributed systems.

Load-bearing premise

Node execution times behave like independent erasures, so that the probability the system finishes equals the probability a codeword is not fully recovered from erasures.

What would settle it

Measure the empirical distribution of node run times for a matrix-multiplication job on real hardware, compute the observed average completion time, and check whether it matches the codeword error rate of the same (n,k) code over an erasure channel with erasure probability taken from the measured run-time statistics.

Figures

Figures reproduced from arXiv: 1906.10105 by Hessam Mahdavifar, Mahdi Soleymani, Mohammad Vahid Jamali.

Figure 1
Figure 1. Figure 1: Scaled average execution time of a homogeneous dis [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
read the original abstract

We consider the problem of coded distributed computing where a large linear computational job, such as a matrix multiplication, is divided into $k$ smaller tasks, encoded using an $(n,k)$ linear code, and performed over $n$ distributed nodes. The goal is to reduce the average execution time of the computational job. We provide a connection between the problem of characterizing the average execution time of a coded distributed computing system and the problem of analyzing the error probability of codes of length $n$ used over erasure channels. Accordingly, we present closed-form expressions for the execution time using binary random linear codes and the best execution time any linear-coded distributed computing system can achieve. It is also shown that there exist good binary linear codes that attain, asymptotically, the best performance any linear code, not necessarily binary, can achieve. We also investigate the performance of coded distributed computing systems using polar and Reed-Muller (RM) codes that can benefit from low-complexity decoding, and superior performance, respectively, as well as explicit constructions. The proposed framework in this paper can enable efficient designs of distributed computing systems given the rich literature in the channel coding theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The paper connects the average execution time of a coded distributed computing system (linear job split into k tasks, encoded with an (n,k) linear code over n nodes) to the word-error probability of the code over a time-dependent erasure channel. It derives closed-form expressions for execution time under binary random linear codes and for the optimal linear code, proves that good binary linear codes achieve the optimal asymptotic performance of any linear code, and evaluates explicit constructions including polar and Reed-Muller codes.

Significance. If the modeling reduction holds, the result supplies exact performance limits and asymptotic optimality statements that directly import known results from erasure-channel coding theory into distributed-computing design. The explicit treatment of random linear, polar, and RM codes supplies both theoretical benchmarks and low-complexity practical options.

minor comments (4)
  1. §II (model): the precise mapping from node completion times to the erasure probability p(t) should be stated as an explicit integral or expectation so that the subsequent closed-form claims can be verified without external reference.
  2. Theorem 1 and Corollary 1: the closed-form expressions for random binary linear codes rely on the rank-deficiency probability of random GF(2) matrices; a short self-contained derivation or citation to the exact formula used would improve readability.
  3. Figure 3 (polar vs. RM comparison): axis labels and legend entries are too small for print; the caption should also state the precise parameters (n,k) and the erasure model used.
  4. Notation: the symbol T_avg is introduced without an explicit definition equation; adding Eq. (X) for the average execution time would eliminate ambiguity when the erasure-channel connection is invoked.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper models node execution times as i.i.d. erasures and equates average completion time to the word-error probability of an (n,k) linear code over a time-dependent BEC. This equivalence is an external modeling assumption, not a self-referential equation. Closed-form expressions for random binary linear codes follow directly from the known rank-deficiency probability of random GF(2) matrices; the asymptotic optimality claim follows from the fact that both binary and non-binary linear codes achieve BEC capacity (vanishing excess nodes in the o(n) limit). No parameters are fitted to the target quantity and then renamed as predictions, no self-citation chain is load-bearing for the central reduction, and no ansatz is imported via prior work by the same authors. The derivation is therefore self-contained against external coding-theory benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard assumptions from coding theory (linearity of codes, memoryless erasure channel) and distributed computing (independent node execution times, linear computational jobs). No free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Node execution times can be modeled as independent erasures whose statistics make average completion time identical to codeword error probability.
    This modeling choice is required for the connection stated in the abstract to hold and for all subsequent closed-form expressions.
  • domain assumption The computational job is linear (e.g., matrix multiplication) so that coded tasks remain valid linear combinations.
    Invoked when the problem is defined as dividing a large linear computational job.

pith-pipeline@v0.9.0 · 5738 in / 1565 out tokens · 26463 ms · 2026-05-25T16:45:42.511741+00:00 · methodology

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Reference graph

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