The $k$-Dimensional Weisfeiler-Leman Algorithm

TL;DR

This paper details the $k$-dimensional Weisfeiler-Leman Algorithm, providing an optimized version with improved runtime complexity of $O(n^{k+1}\log n)$ for fixed $k$, enhancing graph isomorphism testing efficiency.

Contribution

It presents an optimized implementation of the $k$-dimensional Weisfeiler-Leman Algorithm with a significantly improved runtime analysis for fixed $k$.

Findings

Optimized algorithm runs in $O(n^{k+1}\log n)$ time.

Provides detailed analysis from Immerman-Lander (1990).

Enhances efficiency of graph isomorphism testing.

Abstract

In this note, we provide details of the $k$-dimensional Weisfeiler-Leman Algorithm and its analysis from Immerman-Lander (1990). In particular, we present an optimized version of the algorithm that runs in time $O(n^{k+1}\log n)$, where $k$ is fixed (not varying with $n$).