Consider a weighted, undirected graph cellularly embedded on a topological surface. The function assigning to each free homotopy class of closed curves the length of a shortest cycle within this homotopy class is called the marked length spectrum. The (unmarked) length spectrum is obtained by just listing the length values of the marked length spectrum in increasing order. In this paper, we describe algorithms for computing the (un)marked length spectra of graphs embedded on the torus. More specifically, we preprocess a weighted graph of complexity $n$ in time $O(n^2 \log \log n)$ so that, given a cycle with $\ell$ edges representing a free homotopy class, the length of a shortest homotopic cycle can be computed in $O(\ell+\log n)$ time. Moreover, given any positive integer $k$, the first $k$ values of its unmarked length spectrum can be computed in time $O(k \log n)$. Our algorithms are based on a correspondence between weighted graphs on the torus and polyhedral norms. In particular, we give a weight independent bound on the complexity of the unit ball of such norms. As an immediate consequence we can decide if two embedded weighted graphs have the same marked spectrum in polynomial time. We also consider the problem of comparing the unmarked spectra and provide a polynomial time algorithm in the unweighted case and a randomized polynomial time algorithm otherwise.

翻译：考虑拓扑曲面上细胞嵌入的加权无向图。为每条闭曲线的自由同伦类赋予该类中最短环的长度，该函数称为标记长度谱。（未标记的）长度谱则通过按递增顺序列出标记长度谱的长度值获得。本文描述了计算环面上嵌入图的（未）标记长度谱算法。具体而言，我们在 $O(n^2 \log \log n)$ 时间内预处理一个复杂度为 $n$ 的加权图，使得对于任意由 $\ell$ 条边表示自由同伦类的环，可在 $O(\ell+\log n)$ 时间内计算出最短同伦环的长度。此外，给定任意正整数 $k$，可在 $O(k \log n)$ 时间内计算出其未标记长度谱的前 $k$ 个值。我们的算法基于环面上加权图与多面体范数之间的对应关系。特别地，我们给出了此类范数单位球复杂度的与权重的无关上界。作为直接推论，可在多项式时间内判定两个嵌入加权图是否具有相同的标记谱。我们还考虑了比较未标记谱的问题，并在无权情况下给出了多项式时间算法，在其他情况下给出了随机多项式时间算法。