The simultaneous conjugacy problem in the symmetric group

Brodnik, Andrej; Malnič, Aleksander; Požar, Rok

doi:10.1090/mcom/3637

Cited by 4 publications

(11 citation statements)

References 11 publications

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“…There has been some progress in the direction of providing a lower bound. In particular it is known that the communication complexity of the simultaneous conjugation problem is Ω(dn log(n)), for d > 1, and that under the decision tree model the search version of the simultaneous conjugation problem has lower bound of Ω(n log n) [6].…”

Section: Conjecture 12mentioning

confidence: 99%

“…. • The map Aperiodic(M * ) is a homogeneous map if the local type of M is one of the following: (3,3,3,3,6), (3,3,3,4,4). • The map Normalize(Large(Aperiodic(M * ))) is a homogeneous map if the local type of M is (3,3,4,3,4).…”

Section: Torusmentioning

confidence: 99%

“…They admit a simple description in terms of three integer parameters r, s, t where 0 ≤ t < r. The 4-regular quadrangulation Q(r, s, t) is obtained from the (r + 1) × (s + 1) grid with underlying graph P r+1 P s+1 (the Cartesian product of paths on r + 1 and s + 1 vertices) by identifying the "leftmost" path of . Its parameter list is the cyclic sequence of parameters (6,4,2), (12,2,6), (12,2,4), (6,4,4), (12,2,6), (12,2,10). The stabilizers of the automorphism group action are trivial and its full automorphism group is isomorphic to Z 6 × Z 4 .…”

Section: Torusmentioning

confidence: 99%

“…Figure7: The toroidal triangulation T (6, 4, 2). Its parameter list is the cyclic sequence of parameters (6,4,2),(12,2,6),(12,2,4),(6,4,4),(12,2,6),(12,2,10). The stabilizers of the automorphism group action are trivial and its full automorphism group is isomorphic to Z 6 × Z 4 .…”

mentioning

confidence: 99%

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Automorphism groups of maps in linear time

Kawarabayashi,

Mohar,

Nedela

et al. 2020

Preprint

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By a map we mean a 2-cell decomposition of a closed compact surface, i.e., an embedding of a graph such that every face is homeomorphic to an open disc. Automorphism of a map can be thought of as a permutation of the vertices which preserves the vertex-edge-face incidences in the embedding. When the underlying surface is orientable, every automorphism of a map determines an angle-preserving homeomorphism of the surface. While it is conjectured that there is no "truly subquadratic" algorithm for testing map isomorphism for unconstrained genus, we present a linear-time algorithm for computing the generators of the automorphism group of a map, parametrized by the genus of the underlying surface. The algorithm applies a sequence of local reductions and produces a uniform map, while preserving the automorphism group. The automorphism group of the original map can be reconstructed from the automorphism group of the uniform map in linear time. We also extend the algorithm to non-orientable surfaces by making use of the antipodal double-cover.

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Section: Conjecture 12mentioning

confidence: 99%

Section: Torusmentioning

confidence: 99%

Section: Torusmentioning

confidence: 99%

mentioning

confidence: 99%

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Automorphism groups of maps in linear time

Kawarabayashi,

Mohar,

Nedela

et al. 2020

Preprint

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show abstract

“…Later, in 1989 Sridhar presented an

O(dn\mathrm{log}(dn))

‐time algorithm [11]. However, his algorithm does not work correctly as we recently showed in [2].…”

Section: Introductionmentioning

confidence: 99%

A subquadratic algorithm for the simultaneous conjugacy problem

Brodnik

Malnič

Požar

2022

Journal of Graph Theory

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The d $d$‐Simultaneous Conjugacy problem in the symmetric group S n ${S}_{n}$ asks whether there exists a permutation τ ∈ S n $\tau \in {S}_{n}$ such that b j = τ − 1 a j τ ${b}_{j}={\tau }^{-1}{a}_{j}\tau $ holds for all j = 1 , 2 , … , d $j=1,2,\ldots ,d$, where a 1 , a 2 , … , a d ${a}_{1},{a}_{2},\ldots ,{a}_{d}$ and b 1 , b 2 , … , b d ${b}_{1},{b}_{2},\ldots ,{b}_{d}$ are given sequences of permutations in S n ${S}_{n}$. The time complexity of existing algorithms for solving the problem is O ( d n 2 ) $O(d{n}^{2})$. We show that for a given positive integer d $d$ the d $d$‐Simultaneous Conjugacy problem in S n ${S}_{n}$ can be solved in o ( n 2 ) $o({n}^{2})$ time. Our algorithm solves a number of problems from various fields of mathematics and computer science.

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Automorphisms and isomorphisms of maps in linear time

Kawarabayashi,

Mohar,

Nedela

et al. 2024

ACM Trans. Algorithms

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A map is a $2$ -cell decomposition of a closed compact surface, i.e., an embedding of a graph such that every face is homeomorphic to an open disc. An automorphism of a map can be thought of as a permutation of the vertices which preserves the vertex-edge-face incidences in the embedding. Every automorphism of a map determines an angle-preserving homeomorphism of the surface. While it is conjectured that there is no “truly subquadratic” algorithm for testing map isomorphism for unconstrained genus, we present a linear-time algorithm for computing the generators of the automorphism group of a map on an orientable surface of genus $g\neq 0$ , parametrized by the genus $g$ . A map on an orientable surface is uniform if the cyclic vector of sizes of faces incident to a vertex $v$ does not depend on the choice of $v$ . The algorithm applies a sequence of local reductions and produces a uniform map, while preserving the automorphism group. The automorphism group of the original map can be reconstructed from the automorphism group of the associated uniform map in linear time. We also extend the algorithm to non-orientable surfaces by making use of the antipodal double-cover. The algorithm can be used to solve the map isomorphism problem between maps (orientable or non-orientable) of bounded negative Euler characteristic.

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The simultaneous conjugacy problem in the symmetric group

Cited by 4 publications

References 11 publications

Automorphism groups of maps in linear time

Automorphism groups of maps in linear time

A subquadratic algorithm for the simultaneous conjugacy problem

Automorphisms and isomorphisms of maps in linear time

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