Reconstructing Bounded Treelength Graphs with Linearithmic Shortest Path Distance Queries

2026-03-11

Canadian Conference on Computational Geometry

AI Summary

This work addresses the problem of efficiently reconstructing the edge set of a connected graph under the constraint that the only available operation is querying the shortest-path distance between any two vertices. Focusing on graphs with bounded maximum degree Δ and bounded tree-length tl, the paper proposes a novel algorithm that integrates structural properties of such graphs, a deterministic query strategy, and a divide-and-conquer approach. The resulting method achieves a query complexity of \(O_{\Delta,\text{tl}}(n \log n)\), improving upon the previous best-known bound by a factor of \(\log n\). Notably, this matches the information-theoretic lower bound established for bounded chordal graphs, thereby attaining tight query complexity for this general graph class for the first time.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Heuristic Search

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $\Delta$ and bounded treelength $\mathrm{tl}$ connected graphs and show that reconstruction can be done in $O_{\Delta,\mathrm{tl}}(n \log n)$ 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.

Problem

• graph reconstruction
• shortest path distance queries
• bounded treelength
• bounded degree
• query complexity

Innovation

• graph reconstruction
• shortest path distance queries
• bounded treelength
• deterministic algorithm
• query complexity