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Leveraging Structural Knowledge for Solving Election in Anonymous Networks with Shared Randomness

Jérémie Chalopin, Emmanuel Godard

TL;DR

A comprehensive picture illustrating how knowledge influences the computability of the Election problem in arbitrary anonymous graphs with shared randomness is provided.

Abstract

We study the classical Election problem in anonymous net- works, where solutions can rely on the use of random bits, which may be either shared or unshared among nodes. We provide a complete char- acterization of the conditions under which a randomized Election algo- rithm exists, for arbitrary structural knowledge. Our analysis considers both Las Vegas and Monte Carlo randomized algorithms, under the as- sumptions of shared and unshared randomness. In our setting, random sources are considered shared if the output bits are identical across spe- cific subsets of nodes. The algorithms and impossibility proofs are extensions of those of [5] for the deterministic setting. Our results are a complete generalization of those from [8]. Moreover, as applications, we consider many specific knowledge: no knowledge, a bound on the size, a bound on the number of nodes sharing a source, the size, or the full topology of the network. For each of them, we show how the general characterizations apply, showing they actually correspond to classes of structural knowledge. We also de- scribe also how randomized Election algorithms from the literature fits in this landscape. We therefore provide a comprehensive picture illustrating how knowledge influences the computability of the Election problem in arbitrary anonymous graphs with shared randomness.

Leveraging Structural Knowledge for Solving Election in Anonymous Networks with Shared Randomness

TL;DR

A comprehensive picture illustrating how knowledge influences the computability of the Election problem in arbitrary anonymous graphs with shared randomness is provided.

Abstract

We study the classical Election problem in anonymous net- works, where solutions can rely on the use of random bits, which may be either shared or unshared among nodes. We provide a complete char- acterization of the conditions under which a randomized Election algo- rithm exists, for arbitrary structural knowledge. Our analysis considers both Las Vegas and Monte Carlo randomized algorithms, under the as- sumptions of shared and unshared randomness. In our setting, random sources are considered shared if the output bits are identical across spe- cific subsets of nodes. The algorithms and impossibility proofs are extensions of those of [5] for the deterministic setting. Our results are a complete generalization of those from [8]. Moreover, as applications, we consider many specific knowledge: no knowledge, a bound on the size, a bound on the number of nodes sharing a source, the size, or the full topology of the network. For each of them, we show how the general characterizations apply, showing they actually correspond to classes of structural knowledge. We also de- scribe also how randomized Election algorithms from the literature fits in this landscape. We therefore provide a comprehensive picture illustrating how knowledge influences the computability of the Election problem in arbitrary anonymous graphs with shared randomness.
Paper Structure (20 sections, 15 theorems, 2 figures, 2 algorithms)

This paper contains 20 sections, 15 theorems, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Proof Techniques and Contributions.
  3. The Model and Main Technical Results
  4. Main Results
  5. Symmetric Coverings and the Election Problem for a Labeled Graph
  6. Preliminaries
  7. Symmetric Coverings
  8. Election in a $B-$Labeled Graph and Symmetric Coverings
  9. A Las Vegas Election Algorithm knowing the Size
  10. Some Properties of Algorithm ${\mathcal{M}}\xspace$
  11. Quasi-coverings and The Election Problem for a Family of Labeled Graphs
  12. Quasi-Coverings
  13. Las Vegas and Monte Carlo Election in a Family of Labeled Graphs
  14. Applications
  15. Representative Cases of Structural Knowledge
  16. ...and 5 more sections

Key Result

theorem 1

Let ${{\mathcal{F}}\xspace}$ be a recursive family of connected symmetric $B-$labeled digraphs. There exists a Las Vegas Election algorithm for ${{\mathcal{F}}\xspace}$ if and only if every labeled digraphs of ${{\mathcal{F}}\xspace}$ is $B-$minimal, and there exists a recursive function $\tau:{\mat

Figures (2)

  • Figure 1: Quasi-coverings diagram. The ball $B_{{\mathbf D\xspace}\xspace_1}(v_1,r)$ captures "the existence of large enough area of one graph" (${\mathbf D\xspace}\xspace_1$) "that looks locally like another graph" (${\mathbf D\xspace}\xspace_0$).
  • Figure 2: Summary of our Election computability results for $B-$minimal graphs and various knowledge, with previously known results.

Theorems & Definitions (30)

  • theorem 1
  • theorem 2
  • lemma 1: Probabilistic Lifting Lemma
  • proof
  • theorem 3
  • proof : Necessary part
  • proposition 1
  • proof
  • definition 1
  • lemma 2
  • ...and 20 more