# Efficient space virtualisation for Hoshen--Kopelman algorithm

## TL;DR

This paper introduces an efficient space virtualisation technique for the Hoshen--Kopelman algorithm, enabling minimal parallel overhead and application to compute percolation thresholds in four-dimensional lattices with various neighbor configurations.

## Contribution

The paper presents a novel space virtualisation method for the Hoshen--Kopelman algorithm, reducing communication costs and demonstrating its effectiveness in calculating percolation thresholds in complex 4D lattices.

## Key findings

- Minimal parallel overhead observed during computations.
- Accurate percolation thresholds obtained for various neighbor configurations.
- Method applicable to high-dimensional lattice percolation problems.

## Abstract

In this paper the efficient space virtualisation for Hoshen--Kopelman algorithm is presented. We observe minimal parallel overhead during computations, due to negligible communication costs. The proposed algorithm is applied for computation of random-site percolation thresholds for four dimensional simple cubic lattice with sites' neighbourhoods containing next-next-nearest neighbours (3NN). The obtained percolation thresholds are $p_C(\text{NN})=0.19680(23)$, $p_C(\text{2NN})=0.08410(23)$, $p_C(\text{3NN})=0.04540(23)$, $p_C(\text{2NN+NN})=0.06180(23)$, $p_C(\text{3NN+NN})=0.04000(23)$, $p_C(\text{3NN+2NN})=0.03310(23)$, $p_C(\text{3NN+2NN+NN})=0.03190(23)$, where 2NN and NN stand for next-nearest neighbours and nearest neighbours, respectively.