Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Error-resilient Distributed Local Verification

Paweł Garncarek, Tomasz Jurdzinski, Dariusz Kowalski, Subhajit Pramanick

Abstract

We study verification (decision) problems for graph properties in distributed networks under the locally checkable labeling framework, where nodes use labels (proofs) and local neighborhoods to decide acceptance or rejection. Our focus is twofold. First, we study cycle detection. While it is known that this can be verified using 3 labels with access to the 1-hop neighborhood, we introduce a novel gadget that encodes direction along a path using only 2 labels and access to a 3-hop neighborhood. This yields a cycle-detection labeling scheme with just 2 labels and may be of independent interest. Second, we consider adversarially corrupted labelings, where each node has access to a local neighborhood within which a fraction of nodes may receive erroneous labels. We introduce a general algorithmic framework, called refix, that transforms a base verification algorithm for a property P operating on labels within a d-hop neighborhood into one that tolerates up to i erroneous labels within a radius d+2i, by accessing a d+2i-hop neighborhood. We demonstrate applications to cycle detection, cycle absence, and bipartiteness, and provide lower bounds relating the number of errors to the required neighborhood size.

Error-resilient Distributed Local Verification

Abstract

We study verification (decision) problems for graph properties in distributed networks under the locally checkable labeling framework, where nodes use labels (proofs) and local neighborhoods to decide acceptance or rejection. Our focus is twofold. First, we study cycle detection. While it is known that this can be verified using 3 labels with access to the 1-hop neighborhood, we introduce a novel gadget that encodes direction along a path using only 2 labels and access to a 3-hop neighborhood. This yields a cycle-detection labeling scheme with just 2 labels and may be of independent interest. Second, we consider adversarially corrupted labelings, where each node has access to a local neighborhood within which a fraction of nodes may receive erroneous labels. We introduce a general algorithmic framework, called refix, that transforms a base verification algorithm for a property P operating on labels within a d-hop neighborhood into one that tolerates up to i erroneous labels within a radius d+2i, by accessing a d+2i-hop neighborhood. We demonstrate applications to cycle detection, cycle absence, and bipartiteness, and provide lower bounds relating the number of errors to the required neighborhood size.
Paper Structure (17 sections, 7 theorems, 7 figures, 3 tables)

This paper contains 17 sections, 7 theorems, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model and Preliminaries
  3. Verification Problems with Oracular Labelings
  4. Cycle Detection with (Oracular) 3 Labels
  5. Cycle Detection with (Oracular) $n$ Labels
  6. Impossibility for Cycle Detection with 2 Labels and View Distance 1
  7. Cycle Detection with (Oracular) 2 Labels and View Distance 3
  8. Cycle Absence & Bipartiteness with Oracular Labels
  9. Cycle Absence:
  10. Bipartiteness:
  11. Algorithmic Framework for Errors: refix (resilient error fixing) Framework
  12. Verification Problems with Erroneous Labels
  13. Cycle Detection with Erroneous Labeling
  14. Cycle Absence with Erroneous Labeling
  15. Bipartiteness with Erroneous Labeling
  16. ...and 2 more sections

Key Result

Theorem 1

The labeling $L_O$ together with the local verification algorithm $\mathcal{A}\textsc{-3labels}$ constitute a correct locally checkable labeling scheme for cycle detection under a $(L_O, \mathcal{A}\textsc{-3labels})_{3,1}$-system.

Figures (7)

  • Figure 1: An example of the labeling scheme with $3$ labels. The nodes on the cycles and on paths connecting cycles are highlighted within a shaded region. The implied direction from the nodes with labels $0,1$ and $2$ is also marked.
  • Figure 2: $l_1$ and $l_2$ denote the two labels. A blue star ($*$) indicates that the label is copied to the corresponding node below. The left, middle, and right figures correspond to Case (i), Case (ii), and Case (iii), respectively.
  • Figure 3: An example of the labeling scheme with $2$ labels, where every node has the view distance 3.
  • Figure 4: Highlighted route corresponds to tree strings of the form $l_{-2} l_{-1} l_0 l_{1} l_{2}$
  • Figure 5: Highlighted route corresponds to tree string of the form $l_{-2} l_{-1} l_0 l_{-1} l_{-2}$
  • ...and 2 more figures

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1: Imagined Labeling
  • Theorem 5
  • Remark 1
  • Claim 3
  • Claim 4
  • Proposition 1
  • ...and 1 more