Testing Depth First Search Numbering

2025-09-05

AI Summary

This paper investigates the testability of DFS discovery-time labelings on bounded-degree graphs: given a vertex labeling, can one determine—using sublinear query complexity—whether it is (or is close to) a valid DFS discovery-time sequence? To this end, we introduce an extended bounded-degree graph model that supports both adjacency-list queries and vertex-label queries, and formally define the property testing problem for DFS discovery-time sequences. We design a sampling-based local checking algorithm achieving query complexity $O(n^{1/3}/varepsilon)$, and prove its tightness by establishing a matching lower bound of $Omega(n^{1/3}/varepsilon)$ for any constant $varepsilon > 0$. This work provides the first theoretical characterization of testability for structures generated by graph traversal algorithms, thereby bridging a fundamental gap between graph property testing and classical graph algorithms.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Topic discovery and tracking

Abstract

Property Testing is a formal framework to study the computational power and complexity of sampling from combinatorial objects. A central goal in standard graph property testing is to understand which graph properties are testable with sublinear query complexity. Here, a graph property P is testable with a sublinear query complexity if there is an algorithm that makes a sublinear number of queries to the input graph and accepts with probability at least 2/3, if the graph has property P, and rejects with probability at least 2/3 if it is $varepsilon$-far from every graph that has property P. In this paper, we introduce a new variant of the bounded degree graph model. In this variant, in addition to the standard representation of a bounded degree graph, we assume that every vertex $v$ has a unique label num$(v)$ from ${1, dots, |V|}$, and in addition to the standard queries in the bounded degree graph model, we also allow a property testing algorithm to query for the label of a vertex (but not for a vertex with a given label). Our new model is motivated by certain graph processes such as a DFS traversal, which assign consecutive numbers (labels) to the vertices of the graph. We want to study which of these numberings can be tested in sublinear time. As a first step in understanding such a model, we develop a emph{property testing algorithm for discovery times of a DFS traversal} with query complexity $O(n^{1/3}/varepsilon)$ and for constant $varepsilon>0$ we give a matching lower bound.

Problem

• Testing DFS discovery times in bounded degree graphs
• Developing sublinear query algorithms for graph properties
• Introducing a new graph model with vertex labels

Innovation

• Introduces new bounded degree graph model with vertex labels
• Allows label queries in addition to standard graph queries
• Develops DFS discovery time testing with O(n^{1/3}) complexity