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Reconstructing Bounded Treelength Graphs with Linearithmic Shortest Path Distance Queries

Chirag Kaudan, Amir Nayyeri

TL;DR

This result improves over the best known algorithm for this graph class by a $\log n$ factor and matches the known lower bound for the class of graphs with bounded chordality, which is a subclass of bounded treelength graphs.

Abstract

We consider the following graph reconstruction problem: given an unweighted connected graph $G = (V,E)$ with visible vertex set $V$ and an oracle which takes two vertices $u,v \in V$ and returns the shortest path distance between $u$ and $v$, how many queries are needed to reconstruct $E$? Specifically, we consider bounded degree $Δ$ and bounded treelength $\mathrm{tl}$ connected graphs and show that reconstruction can be done in $O_{Δ,\mathrm{tl}}(n \log n)$ queries with a deterministic algorithm. This result improves over the best known algorithm (deterministic or randomized) for this graph class by a $\log n$ factor and matches the known lower bound for the class of graphs with bounded chordality, which is a subclass of bounded treelength graphs.

Reconstructing Bounded Treelength Graphs with Linearithmic Shortest Path Distance Queries

TL;DR

This result improves over the best known algorithm for this graph class by a factor and matches the known lower bound for the class of graphs with bounded chordality, which is a subclass of bounded treelength graphs.

Abstract

We consider the following graph reconstruction problem: given an unweighted connected graph with visible vertex set and an oracle which takes two vertices and returns the shortest path distance between and , how many queries are needed to reconstruct ? Specifically, we consider bounded degree and bounded treelength connected graphs and show that reconstruction can be done in queries with a deterministic algorithm. This result improves over the best known algorithm (deterministic or randomized) for this graph class by a factor and matches the known lower bound for the class of graphs with bounded chordality, which is a subclass of bounded treelength graphs.
Paper Structure (10 sections, 10 theorems, 2 equations, 2 figures)

This paper contains 10 sections, 10 theorems, 2 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Tree decomposition and treelength.
  4. The layering tree.
  5. Reconstruction Algorithm
  6. Extending the layering tree
  7. Extending graph reconstruction
  8. Logarithmic search for the connected component.
  9. Brute force search to find neighbors.
  10. Proof of the main result

Key Result

Theorem 1

Given a hidden graph $G$, and an integer $\tau\geq \mathrm{tl}(G)$, there is an algorithm that reconstructs $G$ using at most $O(\Delta^{3\tau+2} \cdot n \log n)$ SP queries, where $n$ is the number of vertices in $G$ and $\Delta$ is its maximum degree.

Figures (2)

  • Figure 1: Left: the parts of a graph, shaded in purple, based on the BFS layering done with respect to $s$; Right: the layering tree whose vertices correspond to the parts.
  • Figure 2: A shortest $u\text{-}v$ path $\pi$ in $G \setminus L_{\leq k-1}$ and its $x \text{-} y$ subpath $F$ (in red) in the proof of \ref{['lem:ell_Search']}. Parts are represented by the shaded ellipses.

Theorems & Definitions (16)

  • Theorem 1
  • Lemma 1
  • Lemma 2: Dourisbourne and Gavoille journals/dm/DourisboureG07, Theorem 8
  • Lemma 3
  • proof
  • Corollary 1
  • Lemma 4
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 6 more