# Asymptotically faster algorithm for counting self-avoiding walks and self-avoiding polygons

## TL;DR

This paper introduces a significantly faster algorithm for counting self-avoiding walks and polygons on lattices, with improved asymptotic runtime bounds in two and higher dimensions, advancing combinatorial enumeration methods.

## Contribution

The paper presents a new algorithm with asymptotic runtime improvements for counting self-avoiding walks and polygons across various lattice dimensions.

## Key findings

- Runs in time exp(C√(n log n)) for 2D lattices
- Runs in time exp(C_d n^{(d-1)/d} log n) for d-dimensional lattices
- Provides asymptotic bounds that outperform previous methods

## Abstract

We give an algorithm for counting self-avoiding walks or self-avoiding polygons that runs in time $\exp(C\sqrt{n\log n})$ on 2-dimensional lattices and time $\exp(C_dn^{(d-1)/d}\log n)$ on $d$-dimensional lattices for $d>2$.