Quasi-linear distance query reconstruction for graphs of bounded treelength

TL;DR

This work presents an algorithm to reconstruct an n-vertex connected graph G parameterized by maximum degree $\Delta$ and treelength $k$ in O_{k,\Delta}(n \log^2 n)$ queries (in expectation), which is the first algorithm to achieve quasi-linear complexity for this class of graphs.

Abstract

In distance query reconstruction, we wish to reconstruct the edge set of a hidden graph by asking as few distance queries as possible to an oracle. Given two vertices $u$ and $v$, the oracle returns the shortest path distance between $u$ and $v$ in the graph. The length of a tree decomposition is the maximum distance between two vertices contained in the same bag. The treelength of a graph is defined as the minimum length of a tree decomposition of this graph. We present an algorithm to reconstruct an $n$-vertex connected graph $G$ parameterized by maximum degree $Δ$ and treelength $k$ in $O_{k,Δ}(n \log^2 n)$ queries (in expectation). This is the first algorithm to achieve quasi-linear complexity for this class of graphs. The proof goes through a new lemma that could give independent insight on graphs of bounded treelength.

Theorems & Definitions (15)

• Theorem 1.1
• Lemma 1.1
• Lemma 2.1: robertson1986graph
• Theorem 3.1
• Lemma 3.0
• proof
• proof : Proof of Theorem \ref{['thm:randtl']}
• Claim 3.1
• proof
• Claim 3.2