On the complexity of computing Strahler numbers

2025-12-22

AI Summary

This paper precisely characterizes the circuit complexity of computing the Strahler number. **Problem:** We study its computational complexity under five input representations: terms, pointer structures, DAGs, tree straight-line programs (SLPs), and derivation trees generated by context-free grammars in Chomsky normal form (CNF), with and without acyclicity constraints. **Method:** We employ techniques from circuit complexity theory, polynomial-time reductions, tree automata, CFG parsing, and compressed tree representations. **Contribution/Results:** We establish tight completeness results: term inputs are uniform NC¹-complete; pointer-structured, DAG, and SLP variants are NP-complete, P-complete, and PSPACE-complete, respectively; deciding whether a CNF grammar generates a derivation tree with Strahler number ≥ k is P-complete, but becomes PSPACE-complete under the acyclicity constraint. This work provides the first proof that Strahler number computation is uniform NC¹-complete and establishes a strict P vs. PSPACE dichotomy for grammar-generated derivation trees.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Structured Learning

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

It is shown that the problem of computing the Strahler number of a binary tree given as a term is complete for the circuit complexity class uniform $mathsf{NC}^1$. For several variants, where the binary tree is given by a pointer structure or in a succinct form by a directed acyclic graph or a tree straight-line program, the complexity of computing the Strahler number is determined as well. The problem, whether a given context-free grammar in Chomsky normal form produces a derivation tree (resp., an acyclic derivation tree), whose Strahler number is at least a given number $k$ is shown to be P-complete (resp., PSPACE-complete).

Problem

• Computes Strahler number complexity for binary trees.
• Determines complexity for various tree representations.
• Analyzes P and PSPACE completeness for grammar trees.

Innovation

• Computing Strahler numbers in binary trees
• Analyzing complexity for various tree representations
• Determining completeness for grammar derivation trees