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Uncovering a Random Tree

Authors Benjamin Hackl , Alois Panholzer , Stephan Wagner



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Author Details

Benjamin Hackl
  • Uppsala University, Sweden
  • University of Klagenfurt, Austria
Alois Panholzer
  • TU Wien, Austria
Stephan Wagner
  • Uppsala University, Sweden

Cite AsGet BibTex

Benjamin Hackl, Alois Panholzer, and Stephan Wagner. Uncovering a Random Tree. In 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 225, pp. 10:1-10:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.AofA.2022.10

Abstract

We consider the process of uncovering the vertices of a random labeled tree according to their labels. First, a labeled tree with n vertices is generated uniformly at random. Thereafter, the vertices are uncovered one by one, in order of their labels. With each new vertex, all edges to previously uncovered vertices are uncovered as well. In this way, one obtains a growing sequence of forests. Three particular aspects of this process are studied in this extended abstract: first the number of edges, which we prove to converge to a stochastic process akin to a Brownian bridge after appropriate rescaling. Second, the connected component of a fixed vertex, for which different phases are identified and limiting distributions determined in each phase. Lastly, the largest connected component, for which we also observe a phase transition.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Mathematics of computing → Generating functions
Keywords
  • Labeled tree
  • uncover process
  • functional central limit theorem
  • limiting distribution
  • phase transition

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