We revisit the fundamental problem of I/O-efficiently computing $r$-way separators on planar graphs. An $r$-way separator divides a planar graph with $N$ vertices into $O(r)$ regions of size $O(N/r)$ and $O(\sqrt {Nr})$ boundary vertices in total, where boundary vertices are vertices that are adjacent to more than one region. Such separators are used in I/O-efficient solutions to many fundamental problems on planar graphs such as breadth-first search, finding single-source shortest paths, topological sorting, and finding strongly connected components. Our main result is an I/O-efficient sampling-based algorithm that, given a Koebe-embedding of a graph with $N$ vertices and a parameter $r$, computes an $r$-way separator for the graph under certain assumptions on the size of internal memory. Computing a Koebe-embedding of a planar graph is difficult in practice and no known I/O-efficient algorithm currently exists. Therefore, we show how our algorithm can be generalized and applied directly to Delaunay triangulations without relying on a Koebe-embedding. This adaptation can produce many boundary vertices in the worst-case, however, to our knowledge our result is the first to be implemented in practice due to the many non-trivial and complex techniques used in previous results. Furthermore, we show that our algorithm performs well on real-world data and that the number of boundary vertices is small in practice. Motivated by applications in geometric information systems, we show how our algorithm for Delaunay triangulations can be applied to compute the flow accumulation over a terrain, which models how much water flows over the vertices of a terrain. When given an $r$-way separator, our implementation of the algorithm outperforms traditional sweep-line-based algorithms on the publicly available digital elevation model of Denmark.

中文翻译：

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实用的 I/O 高效多路分离器

我们重新审视了 I/O 高效计算平面图上的 $r$-way 分隔符的基本问题。$r$-way 分隔符将具有 $N$ 个顶点的平面图分成大小为 $O(N/r)$ 和 $O(\sqrt {Nr})$ 个边界顶点的 $O(r)$ 区域，其中边界顶点是与多个区域相邻的顶点。此类分隔符用于平面图上许多基本问题的 I/O 高效解决方案，例如广度优先搜索、查找单源最短路径、拓扑排序和查找强连通分量。我们的主要结果是一种基于 I/O 高效采样的算法，该算法给定了具有 $N$ 个顶点和参数 $r$ 的图的 Koebe 嵌入，在某些假设下计算图的 $r$-way 分隔符关于内部存储器的大小。计算平面图的 Koebe 嵌入在实践中很困难，目前还没有已知的 I/O 高效算法。因此，我们展示了如何在不依赖 Koebe 嵌入的情况下将我们的算法推广并直接应用于 Delaunay 三角剖分。这种适应可以在最坏的情况下产生许多边界顶点，但是，据我们所知，由于先前结果中使用了许多重要且复杂的技术，我们的结果是第一个在实践中实现的。此外，我们表明我们的算法在现实世界数据上表现良好，并且在实践中边界顶点的数量很少。受几何信息系统中应用的启发，我们展示了如何应用我们的 Delaunay 三角剖分算法来计算地形上的流量累积，它模拟有多少水流过地形的顶点。当给定 $r$-way 分隔符时，我们在丹麦公开可用的数字高程模型上的算法实现优于传统的基于扫描线的算法。