论文标题
多项式时间有效位置
Polynomial-time efficient position
论文作者
论文摘要
假设$ s $是正复杂性的表面，$ n \ subset S $是$ s $中的大型火车轨道$τ$的领带社区。假设$α$是$ s $的适当沉浸，必不可少的和非外围弧或曲线。我们提出了一种多项式时间算法，在$ s $，$ n $和$α$的情况下，相对于$ n $，同型$α$具有高效位置。 先前在[Takarajima，2000]和[Masurmosherschleimer，2012]中给出了有效位置的证据。在[Takarajima，2000年]中，为在大于或等于两个的属封闭表面上的浸入式曲线提供了有效位置的建设性证明。没有关于隐含算法的复杂性的讨论。在[Masurmosherschleimer，2012]中，证明了在积极复杂性表面上的Birecurrent火车轨迹方面的嵌入曲线的有效位置。隐含算法通过详尽的搜索运行。无法推导时间范围。 我们注意到，在本文中提出的算法和在封闭表面的情况下仔细阅读[Takarajima，2000]的算法是重合的。但是，本文构成的不仅仅是对塔卡拉吉玛的建设性证明的时间复杂性分析。首先，我们更加笼统，因为我们允许具有边界的表面，而Takarajima仅考虑封闭表面。其次，我们的组合设置使用弧形和曲线具有横向自我交流的曲线，而在[Takarajima，2000年]中进行的互补区域的Barycentric细分也强迫非横向自我干预，即使是最初嵌入的曲线。第三，本文中的算法纯粹是根据局部同义提出的，而[Takarajima，2000年]需要半本地的论点。因此，我们可以并且可以为我们的算法提供伪代码，并证明其正确性。
Suppose that $S$ is a surface of positive complexity and $N \subset S$ is a tie neighbourhood of a large train track $τ$ in $S$. Suppose that $α$ is a properly immersed, essential, and non-peripheral arc or curve in $S$. We present a polynomial-time algorithm that, given $S$, $N$, and $α$, homotopes $α$ into efficient position with respect to $N$. Proofs for the existence of efficient position were previously given in [Takarajima,2000] and [MasurMosherSchleimer,2012]. In [Takarajima,2000], a constructive proof for the existence of efficient position is given for immersed curves on closed surfaces of genus greater than or equal to two. There is no discussion of the complexity of the implied algorithm. In [MasurMosherSchleimer,2012], the existence of efficient position is proved for embedded curves with respect to birecurrent train tracks on surfaces of positive complexity. The implied algorithm operates via an exhaustive search. No time bounds can be deduced. We note that the algorithm presented in this thesis and the algorithm suggested by a careful reading of [Takarajima,2000] coincide in the case of closed surfaces. However, this thesis constitutes more than a time-complexity analysis of Takarajima's constructive proof. Firstly, we are more general as we allow surfaces with boundary, whereas Takarajima only considers closed surfaces. Secondly, our combinatorial set-up uses arcs and curves with transverse self-intersection, whereas the barycentric subdivision of complementary regions carried out in [Takarajima,2000] forces non-transverse self-intersection even for curves which are initially embedded. Thirdly, the algorithm in this thesis is formulated purely in terms of local homotopies, whereas [Takarajima,2000] requires semi-local arguments. Thus, we can, and do, give pseudocode for our algorithm as well as prove its correctness.
