# Minimal and Canonical Images

## TL;DR

This paper introduces new algorithms that efficiently find the canonical image of point sets under permutation groups by leveraging orbit structures and subgroup chains, improving over previous methods.

## Contribution

The paper presents a novel family of algorithms that use orbit and subgroup chain structures to reduce search complexity in finding canonical images.

## Key findings

- Algorithms are correct, as proven formally.
- Experimental results show improved efficiency over prior methods.
- Applicable to various permutation groups.

## Abstract

We describe a family of new algorithms for finding the canonical image of a set of points under the action of a permutation group. This family of algorithms makes use of the orbit structure of the group, and a chain of subgroups of the group, to efficiently reduce the amount of search which must be performed to find a canonical image. We present both a formal proof of correctness of our algorithms and experiments on different permutation groups, which compare our algorithms with the previous state of the art.