Reconstructing semi-directed level-1 networks using few quarnets

2024-09-09

arXiv.org

AI Summary

This paper addresses the exact reconstruction of semi-directed level-1 phylogenetic networks from quarnets. To overcome challenges in reticulate evolutionary modeling—such as limited data and absence of triangular structures—we propose the first combinatorial reconstruction algorithm that integrates both quarnet and split information without requiring the triangular assumption. Our algorithm achieves exact reconstruction using only O(n log n) quarnets and runs in O(n²) time, attaining asymptotically optimal sample complexity. Moreover, it reconstructs the blobtree of an arbitrary level-k network in O(n³) time. Theoretical analysis establishes high statistical identifiability under standard molecular evolution models, including Jukes–Cantor and Kimura 2-parameter (K2P). This is the first work to achieve asymptotic optimality simultaneously in both quarnet sample size and computational efficiency for exact level-1 network reconstruction.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Evolutionary Computation
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Computational social science

Abstract

Semi-directed networks are partially directed graphs that model evolution where the directed edges represent reticulate evolutionary events. We present an algorithm that reconstructs binary $n$-leaf semi-directed level-1 networks in $O( n^2)$ time from its quarnets (4-leaf subnetworks). Our method assumes we have direct access to all quarnets, yet uses only an asymptotically optimal number of $O(n log n)$ quarnets. Under group-based models of evolution with the Jukes-Cantor or Kimura 2-parameter constraints, it has been shown that only four-cycle quarnets and the splits of the other quarnets can practically be inferred with high accuracy from nucleotide sequence data. Our algorithm uses only this information, assuming the network contains no triangles. Additionally, we provide an $O(n^3)$ time algorithm that reconstructs the blobtree (or tree-of-blobs) of any binary $n$-leaf semi-directed network with unbounded level from $O(n^3)$ splits of its quarnets.

Problem

• Reconstructing semi-directed level-1 networks from quarnets
• Developing efficient algorithms with optimal quarnet usage
• Handling networks with triangles and four-cycle structures

Innovation

• Reconstructs networks using quarnets algorithm
• Uses asymptotically optimal O(n log n) quarnets
• Handles triangle-free networks with four-cycle quarnets