scholarly journals FAST ALGORITHMIC NIELSEN–THURSTON CLASSIFICATION OF FOUR-STRAND BRAIDS

2012 ◽  
Vol 21 (05) ◽  
pp. 1250043 ◽  
Author(s):  
MATTHIEU CALVEZ ◽  
BERT WIEST
Keyword(s):  
Conjugacy Class ◽  
Polynomial Time ◽  
Word Length ◽  
Conjugacy Problem ◽  
Time Solution

We give an algorithm which decides the Nielsen–Thurston type of a given four-strand braid. The complexity of our algorithm is quadratic with respect to word length. The proof of its validity is based on a result which states that for a reducible 4-braid which is as short as possible within its conjugacy class (short in the sense of Garside), reducing curves surrounding three punctures must be round or almost round. As an application, we give a polynomial time solution to the conjugacy problem for non-pseudo-Anosov four-strand braids.

Classification of finite groups according to their conjugacy class lengths

2019 ◽  
Vol 22 (1) ◽  
pp. 137-156
Author(s):  
Zeinab Foruzanfar ◽  
İsmai̇l Ş. Güloğlu ◽  
Mehdi Rezaei
Keyword(s):  
Conjugacy Class ◽  
Finite Groups

Abstract In this paper, we classify all finite groups satisfying the following property: their conjugacy class lengths are set-wise relatively prime for any six distinct classes.

scholarly journals Time complexity of the conjugacy problem in relatively hyperbolic groups

2015 ◽  
Vol 25 (05) ◽  
pp. 689-723 ◽  
Author(s):  
Inna Bumagin
Keyword(s):  
Polynomial Time ◽  
Positive Curvature ◽  
Nauk Sssr ◽  
Hyperbolic Group ◽  
Conjugacy Problem ◽  
Hyperbolic Groups ◽  
Search Problem ◽  
Relatively Hyperbolic Groups ◽  
Relatively Hyperbolic Group ◽  
Relatively Hyperbolic

If u and v are two conjugate elements of a hyperbolic group then the length of a shortest conjugating element for u and v can be bounded by a linear function of the sum of their lengths, as was proved by Lysenok in [Some algorithmic properties of hyperbolic groups, Izv. Akad. Nauk SSSR Ser. Mat. 53(4) (1989) 814–832, 912]. Bridson and Haefliger showed in [Metrics Spaces of Non-Positive Curvature (Springer-Verlag, Berlin, 1999)] that in a hyperbolic group the conjugacy problem can be solved in polynomial time. We extend these results to relatively hyperbolic groups. In particular, we show that both the conjugacy problem and the conjugacy search problem can be solved in polynomial time in a relatively hyperbolic group, whenever the corresponding problem can be solved in polynomial time in each parabolic subgroup. We also prove that if u and v are two conjugate hyperbolic elements of a relatively hyperbolic group then the length of a shortest conjugating element for u and v is linear in terms of their lengths.

A polynomial time solution for labeling a rectilinear map

1998 ◽  
Vol 65 (4) ◽  
pp. 201-207 ◽  
Author(s):  
Chung Keung Poon ◽  
Zhu Binhai ◽  
Chin Francis
Keyword(s):  
Polynomial Time ◽  
Time Solution

scholarly journals Polynomial time solution to NP-complete problem Hamiltonian cycle (P=NP Proof)

2021 ◽  
Author(s):  
Yasaman KalantarMotamedi
Keyword(s):  
Computer Science ◽  
Polynomial Time ◽  
Hamiltonian Cycle ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Rigorous Proof ◽  
Complete Problem ◽  
Time Solution ◽  
Np Complete

P vs NP is one of the open and most important mathematics/computer science questions that has not been answered since it was raised in 1971 despite its importance and a quest for a solution since 2000. P vs NP is a class of problems that no polynomial time algorithm exists for any. If any of the problems in the class gets solved in polynomial time, all can be solved as the problems are translatable to each other. One of the famous problems of this kind is Hamiltonian cycle. Here we propose a polynomial time algorithm with rigorous proof that it always finds a solution if there exists one. It is expected that this solution would address all problems in the class and have a major impact in diverse fields including computer science, engineering, biology, and cryptography.

scholarly journals Efficient quantum algorithms for the hidden subgroup problem over semi-direct product groups

2007 ◽  
Vol 7 (5&6) ◽  
pp. 559-570
Author(s):  
Y. Inui ◽  
F. Le Gall
Keyword(s):  
Direct Product ◽  
Polynomial Time ◽  
Efficient Algorithm ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Black Box ◽  
Hidden Subgroup Problem ◽  
Time Quantum ◽  
Subgroup Problem

In this paper, we consider the hidden subgroup problem (HSP) over the class of semi-direct product groups $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_q$, for $p$ and $q$ prime. We first present a classification of these groups in five classes. Then, we describe a polynomial-time quantum algorithm solving the HSP over all the groups of one of these classes: the groups of the form $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_p$, where $p$ is an odd prime. Our algorithm works even in the most general case where the group is presented as a black-box group with not necessarily unique encoding. Finally, we extend this result and present an efficient algorithm solving the HSP over the groups $\mathbb{Z}^m_{p^r}\rtimes\mathbb{Z}_p$.

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