On the complexity of computing Strahler numbers

TL;DR

This paper investigates the computational complexity of determining Strahler numbers in various representations of binary trees, establishing their classification within complexity classes such as NC^1, P, and PSPACE.

Contribution

It proves the circuit complexity completeness of computing Strahler numbers for binary trees and analyzes the complexity for different tree representations and related problems.

Findings

Computing Strahler numbers for binary trees is NC^1-complete.

Determining if a derivation tree has a Strahler number at least k is P-complete.

Deciding the Strahler number for acyclic derivation trees is PSPACE-complete.

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).