最小同调基的快速算法
Fast Algorithms for Minimum Homology Basis
AI总结：
本文研究单纯复形的最小同调基问题，提出三种概念简单的随机算法，时间复杂度优于现有确定性算法，并给出了实用实现及与 ShortLoop 的对比。
摘要：
我们研究寻找最小同调基的问题，即在给定单纯复形 K 中生成具有 $\mathbb{Z}_2$ 系数的 $1$ 维同调类的最轻循环集合。这个问题在过去几年中得到了广泛研究。对于一般复形，目前最好的确定性算法由 Dey 等人提出，运行时间为 $O(N m^{ω-1} + n m g)$，其中 $N$ 表示 K 中单纯形的总数，$m$ 表示 K 中边的数量，$n$ 表示 K 中顶点的数量，$g$ 表示 K 的 $1$ 同调群的秩，$ω$ 表示矩阵乘法的指数。在本文中，我们提出了三种概念上简单的随机算法，用于计算一般单纯复形 K 的最小同调基。第一种算法运行时间为 $\tilde{O}(m^ω)$，第二种算法运行时间为 $O(N m^{ω-1})$，第三种算法运行时间为 $\tilde{O}(N^2 g + N m g{^2} + m g{^3})$，当 $g=O(1)$ 时接近二次时间。我们还研究了在具有 $n$ 个顶点和 $m$ 条边的无向图 $G$ 中寻找最小循环基的问题。该问题已知最好的算法运行时间为 $O(m^ω)$。我们的算法具有更简单的高层描述，但代价略高，运行时间为 $\tilde{O}(m^ω)$。我们还提供了计算一般加权复形最小同调基的实用实现。该实现广泛基于本文描述的算法思想，但在使用实用子程序方面有所不同。在这些子程序中，代价较高的步骤使用了并行实现，从而潜在地解决了规模问题。我们将结果与目前已知的最先进实现（ShortLoop）进行了比较。
英文摘要：
We study the problem of finding a minimum homology basis, that is, a lightest set of cycles that generates the $1$-dimensional homology classes with $\mathbb{Z}_2$ coefficients in a given simplicial complex $K$. This problem has been extensively studied in the last few years. For general complexes, the current best deterministic algorithm, by Dey et al., runs in $O(N m^{ω-1} + n m g)$ time, where $N$ denotes the total number of simplices in $K$, $m$ denotes the number of edges in $K$, $n$ denotes the number of vertices in $K$, $g$ denotes the rank of the $1$-homology group of $K$, and $ω$ denotes the exponent of matrix multiplication. In this paper, we present three conceptually simple randomized algorithms that compute a minimum homology basis of a general simplicial complex $K$. The first algorithm runs in $\tilde{O}(m^ω)$ time, the second algorithm runs in $O(N m^{ω-1})$ time and the third algorithm runs in $\tilde{O}(N^2 g + N m g{^2} + m g{^3})$ time which is nearly quadratic time when $g=O(1)$. We also study the problem of finding a minimum cycle basis in an undirected graph $G$ with $n$ vertices and $m$ edges. The best known algorithm for this problem runs in $O(m^ω)$ time. Our algorithm, which has a simpler high-level description, but is slightly more expensive, runs in $\tilde{O}(m^ω)$ time. We also provide a practical implementation of computing the minimum homology basis for general weighted complexes. The implementation is broadly based on the algorithmic ideas described in this paper, differing in its use of practical subroutines. Of these subroutines, the more costly step makes use of a parallel implementation, thus potentially addressing the issue of scale. We compare results against the currently known state of the art implementation (ShortLoop).
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