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How local constraints influence network diameter and applications to LCL generalizations

Nicolas Bousquet, Laurent Feuilloley, Théo Pierron

TL;DR

Several results on the diameter of trees as a function of the number of nodes are established, which have important consequences on the landscape of locally checkable labelings on degree graphs.

Abstract

In this paper, we investigate how local rules enforced at every node can influence the topology of a network. More precisely, we establish several results on the diameter of trees as a function of the number of nodes, as listed below. These results have important consequences on the landscape of locally checkable labelings (LCL) on \emph{unbounded} degree graphs, a case in which our lack of knowledge is in striking contrast with that of \emph{bounded degree graphs}, that has been intensively studied recently. [See paper for full abstract.]

How local constraints influence network diameter and applications to LCL generalizations

TL;DR

Several results on the diameter of trees as a function of the number of nodes are established, which have important consequences on the landscape of locally checkable labelings on degree graphs.

Abstract

In this paper, we investigate how local rules enforced at every node can influence the topology of a network. More precisely, we establish several results on the diameter of trees as a function of the number of nodes, as listed below. These results have important consequences on the landscape of locally checkable labelings (LCL) on \emph{unbounded} degree graphs, a case in which our lack of knowledge is in striking contrast with that of \emph{bounded degree graphs}, that has been intensively studied recently. [See paper for full abstract.]
Paper Structure (11 sections, 17 theorems, 3 figures)

This paper contains 11 sections, 17 theorems, 3 figures.

Table of Contents

  1. Model and definitions
  2. Discussion, results and techniques for general checkers
  3. Discussion, results, and techniques on restricted checkers
  4. Gap results for maximum diameter
  5. Construction of local checkers of prescribed exact diameter for $d \ge 2$
  6. Encoding sequences and locally testable languages.
  7. Constructions reaching the bound $\Theta(S_{c,d}(n))$.
  8. Extension to exact diameter $O(S_{c,d}(n))$
  9. Minimum diameter at distance $1$ -- Proof of Theorem \ref{['thm:radius-1-MIN']}
  10. Constructions
  11. Proof of Theorem \ref{['thm:radius-1-MIN']}

Key Result

Theorem 6

Every local checker in $\mathcal{L}_{c,d}$ has maximum diameter at most $(4d^2+4d+1)\cdot S_{c,d}(n)$ or $\Theta(n)$, where $S_{c,1}(n)= c^2/9$, $S_{c,2}(n)=(cn^c)^{2/(2c+1)}$, $S_{c,3}(n)=36n/\log^2n$ and $S_{c,d}(n)=4n/g_d(\log n)$ if $d>3$.

Figures (3)

  • Figure 1: The tree $T"$ is the graft of $T'$ in $T$ at $(uv,u'v')$.
  • Figure 2: A $3$-rake on $n$ vertices of diameter $\Theta(n^{1/3})$. The deletion of the top path leaves $n^{1/3}$$2$-rakes.
  • Figure 3: A tree $T$ witnessing $(c_1,c_2) <_L (d_1,d_2)$. Vertices are labeled with their name and color.

Theorems & Definitions (25)

  • Definition 1: View of a node
  • Definition 2: Local checker ; $\mathcal{L}_{c,d}$ ; checkability radius
  • Definition 3: Class accepted/recognized by a local checker
  • Definition 4: Generalized-LCL
  • Definition 5: Exact/minimum/maximum diameter of a checker
  • Theorem 6
  • Theorem 7
  • Corollary 7
  • Theorem 8
  • Theorem 9
  • ...and 15 more