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Indirect Coflow Scheduling

Alexander Lindermayr, Kirk Pruhs, Andréa W. Richa, Tegan Wilson

TL;DR

The paper investigates coflow scheduling in reconfigurable networks when many transfers are small by allowing fractional and indirect routing alongside integral matchings. It introduces ALG, a greedy fractional-routing algorithm, and a dual-fitting analysis using Sender/Receiver Bound relaxations to prove a 16-approximation for average completion time. It also analyzes indirect integral routing, delivering a full characterization of worst-case makespan and average completion time across regimes defined by $B$ and $n$, with an $O( ext{log} n)$-approximation in general and matching lower bounds in the uniform-demand case. The results demonstrate that small transfers can be effectively scheduled with indirect routing, achieving near-optimal performance in several regimes and guiding future pursuit of constant-factor guarantees for more cases. Overall, the work advances understanding of coflow scheduling under realistic, small-demand settings and connects fractional/indirect routing with practical approximation guarantees.$

Abstract

We consider routing in reconfigurable networks, which is also known as coflow scheduling in the literature. The algorithmic literature generally (perhaps implicitly) assumes that the amount of data to be transferred is large. Thus the standard way to model a collection of requested data transfers is by an integer demand matrix $D$, where the entry in row $i$ and column $j$ of $D$ is an integer representing the amount of information that the application wants to send from machine/node $i$ to machine/node $j$. A feasible coflow schedule is then a sequence of matchings, which represent the sequence of data transfers that covers $D$. In this work, we investigate coflow scheduling when the size of some of the requested data transfers may be small relative to the amount of data that can be transferred in one round. fractional matchings and/or that employ indirect routing, and compare the relative utility of these options. We design algorithms that perform much better for small demands than the algorithms in the literature that were designed for large data transfers.

Indirect Coflow Scheduling

TL;DR

The paper investigates coflow scheduling in reconfigurable networks when many transfers are small by allowing fractional and indirect routing alongside integral matchings. It introduces ALG, a greedy fractional-routing algorithm, and a dual-fitting analysis using Sender/Receiver Bound relaxations to prove a 16-approximation for average completion time. It also analyzes indirect integral routing, delivering a full characterization of worst-case makespan and average completion time across regimes defined by and , with an -approximation in general and matching lower bounds in the uniform-demand case. The results demonstrate that small transfers can be effectively scheduled with indirect routing, achieving near-optimal performance in several regimes and guiding future pursuit of constant-factor guarantees for more cases. Overall, the work advances understanding of coflow scheduling under realistic, small-demand settings and connects fractional/indirect routing with practical approximation guarantees.$

Abstract

We consider routing in reconfigurable networks, which is also known as coflow scheduling in the literature. The algorithmic literature generally (perhaps implicitly) assumes that the amount of data to be transferred is large. Thus the standard way to model a collection of requested data transfers is by an integer demand matrix , where the entry in row and column of is an integer representing the amount of information that the application wants to send from machine/node to machine/node . A feasible coflow schedule is then a sequence of matchings, which represent the sequence of data transfers that covers . In this work, we investigate coflow scheduling when the size of some of the requested data transfers may be small relative to the amount of data that can be transferred in one round. fractional matchings and/or that employ indirect routing, and compare the relative utility of these options. We design algorithms that perform much better for small demands than the algorithms in the literature that were designed for large data transfers.

Paper Structure

This paper contains 11 sections, 10 theorems, 20 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Summary of Our Results
  4. Related Work
  5. Notation and Formal Definitions
  6. Fractional Matching, Indirect Routing, and Average Completion Time
  7. Algorithm $\bm{{\mathrm{ALG}}}$ Description:
  8. Sender Bound Problem Statement:
  9. Receiver Bound Problem Statement:
  10. Integral Matching and Indirect Routing
  11. Conclusion

Key Result

theorem 1

There is a polynomial-time indirect routing fractional matching algorithm ${\mathrm{ALG}}$ that outputs, for any valid instance, a schedule with an average completion time that is at most 16 times the optimal average completion time. The algorithm only uses direct routing schedules, so the approxima

Figures (2)

  • Figure 1: The data transmissions for nodes in positions $(0, 0)$ and $(1, 1)$ for the 3 by 3 grid formed from $n=9$ nodes. The matchings $M_1$ and $M_2$ belong to Phase 1, and the matchings $M_3$ and $M_4$ belong to Phase 2.
  • Figure 2: The time-expanded routing scheme of a hypercube with dimension $d=2$ and $n=9$.

Theorems & Definitions (18)

  • theorem 1
  • theorem 2
  • corollary 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • proof : Proof of \ref{['thm:main1']}
  • lemma 3
  • proof
  • ...and 8 more