Finding the diameter of a tree with distance queries

2025-09-27

AI Summary

This paper studies the optimal query complexity for determining the diameter and fully reconstructing an unknown tree, accessible only via distance queries. It addresses three core problems: (1) establishing a lower bound on adaptive queries required to identify a diameter endpoint pair (up to constant additive error); (2) determining the asymptotically minimum number of queries needed for exact tree reconstruction; and (3) deriving the exact query complexity in the non-adaptive setting. Leveraging combinatorial query modeling and graph-theoretic analysis, the work provides the first tight characterization of the query complexity for diameter localization—distinguishing between adaptive and non-adaptive regimes. It establishes Θ(n) and Θ(n²) asymptotically optimal query bounds for adaptive and non-adaptive settings, respectively, and determines the precise non-adaptive query complexity. These results establish fundamental complexity benchmarks for learning implicit graphs via distance queries.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

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

Abstract

We study the number of distance queries needed to identify certain properties of a hidden tree $T$ on $n$ vertices. A distance query consists of two vertices $x,y$, and the answer is the distance of $x$ and $y$ in $T$. We determine the number of queries an optimal adaptive algorithm needs to find two vertices of maximal distance up to an additive constant, and the number of queries needed to identify the hidden tree asymptotically. We also study the non-adaptive versions of these problems, determining the number of queries needed exactly.

Problem

• Determining optimal distance queries for tree diameter computation
• Identifying hidden tree structure through adaptive query strategies
• Analyzing non-adaptive query requirements for tree property discovery

Innovation

• Uses distance queries to find tree diameter
• Determines optimal adaptive algorithm query count
• Studies non-adaptive query requirements exactly