Practical I/O-Efficient Multiway Separators

Abstract

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.