We describe an algorithm which, given two essential curves on a surface $S$, computes their distance in the curve graph of $S$, up to multiplicative and additive errors. As an application, we present an algorithm to decide the Nielsen-Thurston type (periodic, reducible, or pseudo-Anosov) of a mapping class of $S$. The novelty of our algorithms lies in the fact that their running time is polynomial in the size of the input and in the complexity of $S$ -- say, its Euler characteristic. This is in contrast with previously known algorithms, which run in polynomial time in the size of the input for any fixed surface $S$.

翻译：我们描述了一种算法，给定曲面 $S$ 上的两条本质曲线，能够计算它们在 $S$ 的曲线复形中的距离，并允许乘法与加法误差。作为应用，我们提出了一种算法，用于判定 $S$ 的映射类属于何种Nielsen-Thurston类型（周期型、可约型或伪Anosov型）。我们的算法的新颖之处在于，其运行时间关于输入规模和曲面 $S$ 的复杂度（例如欧拉特征）呈多项式级增长。这与先前已知的算法形成对比——后者对于任意固定曲面 $S$，其时序仅在输入规模上呈多项式级增长。