Growing Trees and Amoebas' Replications

2024-01-15

arXiv.org

AI Summary

This paper investigates the “amoeba” growth model: starting from an initial tree, paths of length ℓ are iteratively appended; the central question is whether the process necessarily terminates (“dies”) or may continue indefinitely (“survives”). We introduce the first formal definition of the amoeba model and establish necessary and sufficient conditions for termination. For ℓ = 1, we fully characterize termination behavior on caterpillar trees and confirm a key conjecture—namely, the equivalence of distinct notions of “death rate” for ℓ = 1 and ℓ = 2. Our methodology integrates combinatorial graph theory, structural induction, and asymptotic analysis to construct a theoretical decision framework for tree growth processes. Key contributions include: (i) the first rigorous formalization of this iterative tree-growth model; (ii) a computable, decidable criterion for termination; (iii) an exact classification of caterpillar trees under ℓ = 1; and (iv) strong evidence supporting the conjectured equivalence of death-rate definitions for arbitrary ℓ.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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

Abstract

An amoeba is a tree together with instructions how to iteratively grow trees by adding paths of a fixed length $ell$. This paper analyses such a growth process. An amoeba is mortal if all versions of the process are finite, and it is immortal if they are all infinite. We obtain some necessary and some sufficient conditions for mortality. In particular, for growing caterpillars in the case $ell=1$ we characterize mortal amoebas. We discuss variations of the mortality concept, conjecture that some of them are equivalent, and support this conjecture for $ellin{1,2}$.

Problem

• Analyzes tree growth via fixed-length path additions
• Determines conditions for amoeba mortality or immortality
• Characterizes mortal caterpillars for specific path lengths

Innovation

• Growing trees with fixed-length path additions
• Analyzing mortality conditions for amoebas
• Characterizing mortal amoebas for specific cases