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Rate-limited Shuffling for Distributed Computing

Shanuja Sasi, Onur Günlü

TL;DR

This paper studies the shuffling phase in a distributed computing model with rate-limited links between nodes and considers some special cases of the distributed computing problem through two examples for which it is proved that the inner and outer bounds agree, thereby establishing the capacity regions.

Abstract

This paper studies the shuffling phase in a distributed computing model with rate-limited links between nodes. Each node is connected to all other nodes via a noiseless broadcast link with a finite capacity. For this network, the shuffling phase is described as a distributed index-coding problem to extend an outer bound for the latter to the distributed computing problem. An inner bound on the capacity region is also established by using the distributed composite-coding scheme introduced for the distributed index-coding problem. We consider some special cases of the distributed computing problem through two examples for which we prove that the inner and outer bounds agree, thereby establishing the capacity regions. We, then, generalize the special cases to any number of nodes and computation loads under certain constraints.

Rate-limited Shuffling for Distributed Computing

TL;DR

This paper studies the shuffling phase in a distributed computing model with rate-limited links between nodes and considers some special cases of the distributed computing problem through two examples for which it is proved that the inner and outer bounds agree, thereby establishing the capacity regions.

Abstract

This paper studies the shuffling phase in a distributed computing model with rate-limited links between nodes. Each node is connected to all other nodes via a noiseless broadcast link with a finite capacity. For this network, the shuffling phase is described as a distributed index-coding problem to extend an outer bound for the latter to the distributed computing problem. An inner bound on the capacity region is also established by using the distributed composite-coding scheme introduced for the distributed index-coding problem. We consider some special cases of the distributed computing problem through two examples for which we prove that the inner and outer bounds agree, thereby establishing the capacity regions. We, then, generalize the special cases to any number of nodes and computation loads under certain constraints.
Paper Structure (8 sections, 1 theorem, 19 equations, 2 figures)

This paper contains 8 sections, 1 theorem, 19 equations, 2 figures.

Table of Contents

  1. introduction
  2. Background and Preliminaries
  3. Distributed Index-coding Problem
  4. Multi-sender Unicast Index-Coding Problem
  5. Problem Definition
  6. Outer Bound
  7. Inner Bound
  8. Capacity Regions for Special DC Models

Key Result

Proposition 1

For a DC problem represented by digraph $\mathcal{G}$, if the rate tuple $(R_{(k,f)}: (k,f) \in \mathcal{V})$ is achievable for a given link-capacity tuple $(C_k: k \in [0,K))$, it must satisfy for all $S \subseteq \mathcal{V}$ for which the subgraph of $\mathcal{G}$ induced by $S$ does not contain a directed cycle. ∎

Figures (2)

  • Figure 1: Digraph $\mathcal{G}_1$ corresponding to Example 1.
  • Figure 2: Digraph $\mathcal{G}_2$ corresponding to Example 2.

Theorems & Definitions (5)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Example 1
  • Example 2