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Testing Depth First Search Numbering

Authors Artur Czumaj , Christian Sohler , Stefan Walzer

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Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Randomized Algorithms
  • Graph Algorithms
  • Property Testing

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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 ε-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, … , |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 property testing algorithm for discovery times of a DFS traversal with query complexity O(n^{1/3}/ε) and for constant ε > 0 we give a matching lower bound.

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Artur Czumaj, Christian Sohler, and Stefan Walzer. Testing Depth First Search Numbering. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 78:1-78:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.78

Author Details

Artur Czumaj
  • Department of Computer Science and DIMAP, University of Warwick, Coventry, UK
  • University of Cologne, Germany
Christian Sohler
  • University of Cologne, Germany
Stefan Walzer
  • Karlsruhe Institute of Technology, Germany

Funding

  • Czumaj, Artur: Research partially supported by the Centre for Discrete Mathematics and its Applications (DIMAP), by a Weizmann-UK Making Connections Grant, and by an IBM Award.
  • Walzer, Stefan: Research funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 465963632.

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