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Necessity of Cooperative Transmissions for Wireless MapReduce

Yue Bi, Michèle Wigger

TL;DR

The paper addresses the wireless MapReduce setting, aiming to minimize the Normalized Delivery Time (NDT) for a given computation load $r$. It introduces an IA+ZF scheme that enables $r$-fold transmitter cooperation, achieving new upper bounds on the NDT that improve over prior results for $r=\lfloor (K-1)/2\rfloor$, with explicit DoF analyses for both even and odd $K$. A complementary non-cooperative converse shows that any scheme without transmitter cooperation cannot reach the same NDT, establishing the necessity of cooperation for optimal performance. The authors provide analytic proofs for $K=5$ and numerical validation up to $K=15$, demonstrating the suboptimality of non-cooperative schemes and the advantage of cooperative interference management. Overall, the work highlights the critical role of cooperation in wireless MapReduce and advances the understanding of the fundamental NDT–computation tradeoff in multiuser distributed computing systems.

Abstract

The paper presents an improved upper bound (achievability result) on the optimal tradeoff between Normalized Delivery Time (NDT) and computation load for distributed computing MapReduce systems in certain ranges of the parameters. The upper bound is based on interference alignment combined with zero-forcing. The paper further provides a lower bound (converse) on the optimal NDT-computation tradeoff that can be achieved when IVAs are partitioned into sub-IVAs, and these sub-IVAs are then transmitted (in an arbitrary form) by a single node, without cooperation among nodes. For appropriate linear functions (e.g., XORs), such non-cooperative schemes can achieve some of the best NDT-computation tradeoff points so far obtained in the literature. However, as our lower bound shows, any non-cooperative scheme achieves a worse NDT-computation tradeoff than our new proposed scheme for certain parameters, thus proving the necessity of cooperative schemes like zero-forcing to attain the optimal NDT-computation tradeoff.

Necessity of Cooperative Transmissions for Wireless MapReduce

TL;DR

The paper addresses the wireless MapReduce setting, aiming to minimize the Normalized Delivery Time (NDT) for a given computation load . It introduces an IA+ZF scheme that enables -fold transmitter cooperation, achieving new upper bounds on the NDT that improve over prior results for , with explicit DoF analyses for both even and odd . A complementary non-cooperative converse shows that any scheme without transmitter cooperation cannot reach the same NDT, establishing the necessity of cooperation for optimal performance. The authors provide analytic proofs for and numerical validation up to , demonstrating the suboptimality of non-cooperative schemes and the advantage of cooperative interference management. Overall, the work highlights the critical role of cooperation in wireless MapReduce and advances the understanding of the fundamental NDT–computation tradeoff in multiuser distributed computing systems.

Abstract

The paper presents an improved upper bound (achievability result) on the optimal tradeoff between Normalized Delivery Time (NDT) and computation load for distributed computing MapReduce systems in certain ranges of the parameters. The upper bound is based on interference alignment combined with zero-forcing. The paper further provides a lower bound (converse) on the optimal NDT-computation tradeoff that can be achieved when IVAs are partitioned into sub-IVAs, and these sub-IVAs are then transmitted (in an arbitrary form) by a single node, without cooperation among nodes. For appropriate linear functions (e.g., XORs), such non-cooperative schemes can achieve some of the best NDT-computation tradeoff points so far obtained in the literature. However, as our lower bound shows, any non-cooperative scheme achieves a worse NDT-computation tradeoff than our new proposed scheme for certain parameters, thus proving the necessity of cooperative schemes like zero-forcing to attain the optimal NDT-computation tradeoff.
Paper Structure (15 sections, 5 theorems, 62 equations, 1 figure, 2 tables, 1 algorithm)

This paper contains 15 sections, 5 theorems, 62 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Wireless MapReduce Framework
  3. Map phase
  4. Shuffle phase
  5. Reduce phase
  6. Relation with $\sf r$-fold cooperative channel
  7. Main results
  8. The New IA & ZF Scheme
  9. A scheme for $\sf r = \lfloor (\sf K-1)/2\rfloor$
  10. Analysis of SDoF
  11. Independence of channel coefficients for $\sf K=5$
  12. Encoding
  13. Independence and Calculation of Jacobian matrix
  14. Proof of the Converse Result (Theorem \ref{['thm:split_converse']})
  15. Conclusion

Key Result

Lemma 1

Under the combinatorial file assignment, for any $\sf r\in[\sf K]$,

Figures (1)

  • Figure 1: Comparison of the achieved NDT--computation tradeoff between the proposed scheme and the schemes in li_wireless_2019bi_dof_2022bi_normalized_2024.

Theorems & Definitions (6)

  • Lemma 1
  • Theorem 1: Achievable NDT-computation tradeoff
  • Theorem 2: Converse under Non-Cooperative Transmissions
  • Corollary 1
  • Lemma 2
  • proof