单形的同调基标注及其应用
Annotating Simplices with a Homology Basis and Its Applications
摘要：
设$K$是一个单纯复形，$g$是它的第$p$个同调群$H_p(K)$的秩，它由$Z_2$系数定义.我们表明，我们可以计算一个基础$H$的$H_p(K)$和注释每个$p$ -单形的$K$与二进制向量的长度$g$具有以下属性：注释，总结了所有$p$ -单形在任何$p$ -循环$z$，提供了坐标向量的同源类$[z]$的基础$H$。所有单形的基和注释可以在$O(n^ω)$时间内计算，其中$n$是$K$的大小，$ω<2.376$是一个量，使得两个$n\times n$矩阵可以在$O(n^ω)$时间内相乘。预计算的注释允许回答查询的独立性或$p$ -循环的平凡有效。利用2-复形中边的注释，我们得到了计算1维同调中最优基和最优同调圈的更好算法。具体来说，计算$H_1(K)$的最佳基础，我们提高了时间复杂度已知的问题从$O(n^4)$到$O(n^ω+n^2g^{ω-1})$。这里，$n$表示$K$的2-骨架的大小，$g$表示$H_1(K)$的秩。即使对于曲面，计算与给定的1-循环同源的最优循环也是NP困难的，并且对于曲面，已知花费$2^{O(g)}n\log n$时间的算法。我们扩展此算法的工作与任意2-复合物在$O(n^ω)+2^{O(g)}n^2\log n$时间使用注释。
英文摘要：
Let $K$ be a simplicial complex and $g$ the rank of its $p$-th homology group $H_p(K)$ defined with $Z_2$ coefficients. We show that we can compute a basis $H$ of $H_p(K)$ and annotate each $p$-simplex of $K$ with a binary vector of length $g$ with the following property: the annotations, summed over all $p$-simplices in any $p$-cycle $z$, provide the coordinate vector of the homology class $[z]$ in the basis $H$. The basis and the annotations for all simplices can be computed in $O(n^ω)$ time, where $n$ is the size of $K$ and $ω<2.376$ is a quantity so that two $n\times n$ matrices can be multiplied in $O(n^ω)$ time. The pre-computation of annotations permits answering queries about the independence or the triviality of $p$-cycles efficiently. Using annotations of edges in 2-complexes, we derive better algorithms for computing optimal basis and optimal homologous cycles in 1-dimensional homology. Specifically, for computing an optimal basis of $H_1(K)$, we improve the time complexity known for the problem from $O(n^4)$ to $O(n^ω+n^2g^{ω-1})$. Here $n$ denotes the size of the 2-skeleton of $K$ and $g$ the rank of $H_1(K)$. Computing an optimal cycle homologous to a given 1-cycle is NP-hard even for surfaces and an algorithm taking $2^{O(g)}n\log n$ time is known for surfaces. We extend this algorithm to work with arbitrary 2-complexes in $O(n^ω)+2^{O(g)}n^2\log n$ time using annotations.
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