While algorithms for planar graphs have received a lot of attention, few papers have focused on the additional power that one gets from assuming an embedding of the graph is available. While in the classic sequential setting, this assumption gives no additional power (as a planar graph can be embedded in linear time), we show that this is far from being the case in other settings. We assume that the embedding is straight-line, but our methods also generalize to non-straight-line embeddings. Specifically, we focus on sublinear-time computation and massively parallel computation (MPC). Our main technical contribution is a sublinear-time algorithm for computing a relaxed version of an $r$-division. We then show how this can be used to estimate Lipschitz additive graph parameters. This includes, for example, the maximum matching, maximum independent set, or the minimum dominating set. We also show how this can be used to solve some property testing problems with respect to the vertex edit distance. In the second part of our paper, we show an MPC algorithm that computes an $r$-division of the input graph. We show how this can be used to solve various classical graph problems with space per machine of $O(n^{2/3+\epsilon})$ for some $\epsilon>0$, and while performing $O(1)$ rounds. This includes for example approximate shortest paths or the minimum spanning tree. Our results also imply an improved MPC algorithm for Euclidean minimum spanning tree.
翻译:虽然平面图的算法受到了很多关注,但很少的文件关注了假设嵌入图而获得的额外力量。在经典的顺序设置中,这一假设没有带来额外的力量(因为平面图可以嵌入线性时间),但我们显示,这远远没有在其他设置中出现这种情况。我们假设嵌入是直线的,但我们的方法也一般化为非直线嵌入。具体地说,我们侧重于亚线时间计算和大规模平行计算(MPC)。我们的主要技术贡献是计算一个宽松的美元平面图版本的亚线性时间算法。我们随后展示了如何使用这一算法来估计利普西茨添加图参数的参数(因为平面图可以嵌入最大匹配、最大独立设置或最小占位值设置。我们还展示了如何用这个算法来解决与顶值平面电量值的距离有关的某些属性测试问题。在我们的论文的第二部分中,我们展示了用于计算美元平面平面平面平面平面平面平面平面平面平面图中的某种美元平面算法。我们用这个平面平面平面平面平面平面平面平面,我们展示了这个平面平面平面平面平面平面平面图,我们用这个平面图,用这个平面图。我们展示了这个平面图。