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Constant Degree Networks for Almost-Everywhere Reliable Transmission

Mitali Bafna, Dor Minzer

TL;DR

This work shows a construction of constant-degree networks with efficient protocols that can tolerate a constant fraction of adversarial faults, thus solving the main open problem of Dwork et al.

Abstract

In the almost-everywhere reliable message transmission problem, introduced by [Dwork, Pippenger, Peleg, Upfal'86], the goal is to design a sparse communication network $G$ that supports efficient, fault-tolerant protocols for interactions between all node pairs. By fault-tolerant, we mean that that even if an adversary corrupts a small fraction of vertices in $G$, then all but a small fraction of vertices can still communicate perfectly via the constructed protocols. Being successful to do so allows one to simulate, on a sparse graph, any fault-tolerant distributed computing task and secure multi-party computation protocols built for a complete network, with only minimal overhead in efficiency. Previous works on this problem achieved either constant-degree networks tolerating $o(1)$ faults, constant-degree networks tolerating a constant fraction of faults via inefficient protocols (exponential work complexity), or poly-logarithmic degree networks tolerating a constant fraction of faults. We show a construction of constant-degree networks with efficient protocols (i.e., with polylogarithmic work complexity) that can tolerate a constant fraction of adversarial faults, thus solving the main open problem of Dwork et al.. Our main contribution is a composition technique for communication networks, based on graph products. Our technique combines two networks tolerant to adversarial edge-faults to construct a network with a smaller degree while maintaining efficiency and fault-tolerance. We apply this composition result multiple times, using the polylogarithmic-degree edge-fault tolerant networks constructed in a recent work of [Bafna, Minzer, Vyas'24] (that are based on high-dimensional expanders) with itself, and then with the constant-degree networks (albeit with inefficient protocols) of [Upfal'92].

Constant Degree Networks for Almost-Everywhere Reliable Transmission

TL;DR

This work shows a construction of constant-degree networks with efficient protocols that can tolerate a constant fraction of adversarial faults, thus solving the main open problem of Dwork et al.

Abstract

In the almost-everywhere reliable message transmission problem, introduced by [Dwork, Pippenger, Peleg, Upfal'86], the goal is to design a sparse communication network that supports efficient, fault-tolerant protocols for interactions between all node pairs. By fault-tolerant, we mean that that even if an adversary corrupts a small fraction of vertices in , then all but a small fraction of vertices can still communicate perfectly via the constructed protocols. Being successful to do so allows one to simulate, on a sparse graph, any fault-tolerant distributed computing task and secure multi-party computation protocols built for a complete network, with only minimal overhead in efficiency. Previous works on this problem achieved either constant-degree networks tolerating faults, constant-degree networks tolerating a constant fraction of faults via inefficient protocols (exponential work complexity), or poly-logarithmic degree networks tolerating a constant fraction of faults. We show a construction of constant-degree networks with efficient protocols (i.e., with polylogarithmic work complexity) that can tolerate a constant fraction of adversarial faults, thus solving the main open problem of Dwork et al.. Our main contribution is a composition technique for communication networks, based on graph products. Our technique combines two networks tolerant to adversarial edge-faults to construct a network with a smaller degree while maintaining efficiency and fault-tolerance. We apply this composition result multiple times, using the polylogarithmic-degree edge-fault tolerant networks constructed in a recent work of [Bafna, Minzer, Vyas'24] (that are based on high-dimensional expanders) with itself, and then with the constant-degree networks (albeit with inefficient protocols) of [Upfal'92].
Paper Structure (24 sections, 9 theorems, 10 equations)

This paper contains 24 sections, 9 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Our Results
  3. Implications
  4. Techniques
  5. The Balanced Replacement Product
  6. Routing Protocols on the Composed Graph
  7. Using Composition to Obtain Sparse Networks
  8. Preliminaries
  9. Notations:
  10. Routing Protocols under Adversarial Corruptions
  11. Composition of Protocols using the Replacement Product
  12. The Protocol on $Z$
  13. Analysis of Protocols Generated by \ref{['algo:zz-protocol']}
  14. The outer protocol over clouds:
  15. The inner protocol inside clouds:
  16. ...and 9 more sections

Key Result

Theorem 1.1

There exists $D\in \mathbb{N}$ such that for all ${\varepsilon}>0$ and large enough $n$, there exists a $D$-regular graph $G$ with $\Theta(n)$ vertices and a set of protocols $\mathcal{R}=\{R(u,v)\}_{u,v\in G}$ between all pairs of vertices in $G$, with work complexity $\sf{polylog} n$ and round com

Theorems & Definitions (20)

  • Theorem 1.1
  • Lemma 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Permutation Model
  • Lemma 3.1
  • Claim 3.2
  • proof
  • Claim 3.3
  • ...and 10 more