发表机构
University of Konstanz; ENSEIRB, Bordeaux（康斯坦茨大学; 波尔多ENSEIRB）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究几何图中 $c$-packedness 与 $\lambda$-low-density 参数，提出可扩展计算方法，证明 $c\in O(\lambda\sqrt n)$，并改进平衡分隔符算法，实现 $O(c)$ 查询时间的精确距离预言，实验验证于大型道路网络。
AI 中文摘要
在设计几何图算法时，利用结构参数可以显著改进算法界限。在此背景下，两个突出的参数是 $c$-packedness 和 $\lambda$-low density，两者都局部限制图的复杂度。基于这些参数的参数化算法已被开发用于计算良分离对分解、平衡分隔符以及距离预言。然而，由 $c$ 或 $\lambda$ 参数化的算法的实际适用性仍不清楚。虽然 $c$-packed 和 $\lambda$-low-density 图已被提议作为道路网络的现实模型，但大型真实世界实例的实际参数值至今未知，且现有的理论保证部分过于宽松，难以实际应用。在本文中，我们首先设计了用于近似计算 $c$ 和精确计算 $\lambda$ 的可扩展实现。我们在具有数百万条边的道路网络上的实验揭示了这两个参数之间的显著差距。在理论方面，我们证明了 $c\in O(\lambda\sqrt n)$，这补充了已知结果 $\lambda\in O(c)$。此外，我们提出了改进的平衡分隔符计算参数化算法，在理论和实践中都减小了分隔符大小。我们还展示了如何在多项式时间内计算宽度与相应参数化平衡分隔符大小线性相关的树分解。这一结构结果产生了多种新的算法推论。其中包括在多项式时间预处理后，对 $c$-packed 图具有查询时间 $O(c)$ 的精确距离预言，这改进了先前的 $O(c\log n)$ 界限。我们的实验表明，所提出的技术能高效地产生小的平衡分隔符，并能在大型道路网络上构建简洁的精确距离预言。
英文摘要
When designing algorithms for geometric graphs, exploiting structural parameters can lead to significantly improved bounds. Two prominent parameters in this context are $c$-packedness and $λ$-low density, both of which locally restrict graph complexity. Parameterized algorithms based on these parameters have been developed for computing well-separated pair decompositions, balanced separators, as well as distance oracles. Nevertheless the practical applicability of algorithms parameterized by $c$ or $λ$ remains unclear. While $c$-packed and $λ$-low-density graphs have been proposed as realistic models for road networks, the actual parameter values of large real-world instances have so far remained unknown, and existing theoretical guarantees are partially too loose for practical usage. In this paper we first devise scalable implementations for the approximate computation of $c$ and the exact computation of $λ$. Our experiments on road networks with millions of edges reveals a significant gap between the two parameters. On the theoretical side we prove that $c\in O(λ\sqrt n)$ which complements the known result that $λ\in O(c)$. Furthermore we present improved parameterized algorithms for balanced separator computation that reduce the separator size in theory and practice. We also show how to compute a tree decomposition with a width linear in the respective parameterized balanced separator size in polynomial time. This structural result yields a variety of new algorithmic consequences. Among them is an exact distance oracle with query time $O(c)$ for $c$-packed graphs after polynomial-time preprocessing, which improves upon the previous $O(c\log n)$ bound. Our experiments show that the proposed techniques efficiently produce small balanced separators and enable the construction of concise exact distance oracles on large road networks.
