Revisiting Tree Canonization using polynomials

TL;DR

This paper introduces a new, simple logspace algorithm for tree canonization that uses polynomial representations, offering an alternative to Lindell's traditional approach and extending to other graph classes.

Contribution

It presents a novel logspace algorithm for tree canonization based on polynomial computations, differing from prior methods and leveraging Eisenstein's criterion.

Findings

The algorithm computes a polynomial canon for trees in logspace.

It simplifies the process by avoiding complex recursion and case analysis.

The approach extends to other graph classes.

Abstract

Graph Isomorphism (GI) is a fundamental algorithmic problem. Amongst graph classes for which the computational complexity of GI has been resolved, trees are arguably the most fundamental. Tree Isomorphism is complete for deterministic logspace, a tiny subclass of polynomial time, by Lindell's result. Over three decades ago, he devised a deterministic logspace algorithm that computes a string which is a canon for the input tree -- two trees are isomorphic precisely when their canons are identical. Inspired by Miller-Reif's reduction of Tree Isomorphism to Polynomial Identity Testing, we present a new logspace algorithm for tree canonization fundamentally different from Lindell's algorithm. Our algorithm computes a univariate polynomial as canon for an input tree, based on the classical Eisenstein's criterion for the irreducibility of univariate polynomials. This can be implemented in logspace by invoking the well known Buss et al. algorithm for arithmetic formula evaluation. However, we have included in the appendix a simpler self-contained proof showing that arithmetic formula evaluation is in logspace. This algorithm is conceptually very simple, avoiding the delicate case analysis and complex recursion that constitute the core of Lindell's algorithm. We illustrate the adaptability of our algorithm by extending it to a couple of other classes of graphs.