The quest for efficient sorting is ongoing, and we will explore a graph-based stable sorting strategy, in particular employing comparison graphs. We use the topological sort to map the comparison graph to a linear domain, and we can manipulate our graph such that the resulting topological sort is the sorted array. By taking advantage of the many relations between Hamiltonian paths and topological sorts in comparison graphs, we design a Divide-and-Conquer algorithm that runs in the optimal $O(n \log n)$ time. In the process, we construct a new merge process for graphs with relevant invariant properties for our use. Furthermore, this method is more space-efficient than the famous {\sc MergeSort} since we modify our fixed graph only.

中文翻译：

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使用对应比较图的拓扑排序对数组进行排序

对高效排序的追求正在进行中，我们将探索基于图的稳定排序策略，特别是使用比较图。我们使用拓扑排序将比较图映射到线性域，并且我们可以操作我们的图，使得结果拓扑排序是排序数组。通过利用比较图中哈密顿路径和拓扑排序之间的许多关系，我们设计了一个分治算法，该算法在最优 $O(n\logn)$ 时间内运行。在这个过程中，我们为具有相关不变属性的图构建了一个新的合并过程供我们使用。此外，这种方法比著名的 {\sc MergeSort} 更节省空间，因为我们只修改我们的固定图。