Optimal root recovery for uniform attachment trees and $d$-regular growing trees (2411.18614v1)

Abstract: We consider root-finding algorithms for random rooted trees grown by uniform attachment. Given an unlabeled copy of the tree and a target accuracy $\varepsilon > 0$, such an algorithm outputs a set of nodes that contains the root with probability at least $1 - \varepsilon$. We prove that, for the optimal algorithm, an output set of size $\exp(O(\log^{1/2}(1/\varepsilon)))$ suffices; this bound is sharp and answers a question of Bubeck, Devroye and Lugosi (2017). We prove similar bounds for random regular trees that grow by uniform attachment, strengthening a result of Khim and Loh (2017).

• The paper presents optimal algorithms that balance error probability and output set size for accurately finding the root in uniform attachment trees.
• It extends these methods to d-regular growing trees, achieving similar tight bounds through advanced probabilistic and combinatorial techniques.
• The study advances network archaeology by providing rigorous theoretical insights with practical implications for effective network and phylogenetic analysis.

An Overview of "Optimal Root Recovery for Uniform Attachment Trees and $d$-Regular Growing Trees"

The paper, "Optimal Root Recovery for Uniform Attachment Trees and $d$-Regular Growing Trees," investigates the problem of root reconstruction in two types of random trees: uniform attachment trees and $d$-regular trees. This research is positioned within the broader context of network archaeology and rumor centrality, focusing specifically on developing algorithms that can identify the root of a tree with high confidence while minimizing the size of the node set returned by the algorithm. The authors provide both a theoretical underpinning of the problem and concrete algorithmic solutions, drawing connections to probabilistic methods and combinatorial properties of trees.

Main Contributions

The paper's primary contributions revolve around the development of effective root-finding algorithms for these tree models, with the algorithms aiming to balance between the probability of successfully identifying the root and the computational complexity expressed in the size of the output node set. Notably, the paper addresses a question previously posed in the literature regarding the optimal size of such node sets for given error tolerances.

1. Root-Finding Algorithms: The authors present a root-finding algorithm that improves upon existing methods by providing an optimal balance between the error probability (denoted by $\varepsilon$) and the size of the output node set. Specifically, for uniform attachment trees, the algorithm outputs a set containing the root with probability at least $1 - \varepsilon$. The size of this set is shown to be $\exp(O(\sqrt{\log(1/\varepsilon)}))$, which is proven to be the best possible bound up to a constant. This result solves an open problem by showing the optimality of the previously established lower bounds.
2. Extension to $d$-Regular Trees: The paper extends these results to $d$-regular trees, where each non-leaf node has exactly $d$ neighbors. This generalization requires non-trivial modifications to the techniques used for uniform attachment trees. The bounds here are shown to be similar to those for the uniform attachment trees, reinforcing the robustness of the proposed methods across different types of growing tree models.
3. Probabilistic Analysis: A substantial part of the paper is dedicated to rigorous probabilistic analysis, which is essential in establishing the upper and lower bounds for the number of nodes in the output set. Through a detailed examination of geometrically decaying properties and the application of probabilistic tools such as oligarchy Pólya urn schemes, the authors derive crucial insights into the underlying structure of these trees.

Theoretical and Practical Implications

The theoretical implications of this work are significant, as the results provide a clearer understanding of the limitations and capabilities of root-finding algorithms in stochastic tree models. By proving the optimality of the bounds achieved by their algorithms, the authors lay the groundwork for potential future studies in related models, such as preferential attachment trees or trees subject to different attachment rules.

From a practical perspective, the ability to accurately and efficiently identify the root of a tree has direct applications in areas such as phylogenetics, network forensics, and the reconstruction of network history after events such as information dispersal. The methods proposed could be adapted or extended to a variety of network settings, providing robust tools for practitioners dealing with real-world network analysis problems.

Conclusion and Future Directions

The paper not only contributes fundamental theoretical advancements in the understanding of tree-based network models but also provides practical algorithmic solutions with potential applications in diverse fields. Future research could explore the extension of these methods to more general graph structures, assess the impact of different types of noise and error in observed data, or explore dynamic networks where tree structures evolve over time. The established tight bounds will serve as a benchmark for evaluating new methods in these extended domains.