A subquadratic algorithm for the simultaneous conjugacy problem

TL;DR

This paper presents a novel algorithm that solves the $d$-Simultaneous Conjugacy problem in the symmetric group $S_n$ more efficiently, achieving subquadratic time complexity, which improves upon the previous $O(dn^2)$ algorithms.

Contribution

The authors develop a subquadratic time algorithm for the $d$-Simultaneous Conjugacy problem in $S_n$, reducing the complexity to $o(n^2)$ for fixed $d$.

Findings

The new algorithm operates in subquadratic time for fixed $d$.

It significantly improves the efficiency over previous methods.

The approach is applicable to permutation groups in computational group theory.

Abstract

The $d$-Simultaneous Conjugacy problem in the symmetric group $S_n$ asks whether there exists a permutation $\tau \in S_n$ such that $b_j = \tau^{-1}a_j \tau$ holds for all $j = 1,2,\ldots, d$, where $a_1, a_2,\ldots , a_d$ and $b_1, b_2,\ldots , b_d$ are given sequences of permutations in $S_n$. The time complexity of existing algorithms for solving the problem is $O(dn^2)$. We show that for a given positive integer $d$ the $d$-Simultaneous Conjugacy problem in $S_n$ can be solved in $o(n^2)$ time.