边修改对直径计算及更多问题的代价
The Cost of Changing Edges for Diameter Computation and More
浏览论文内容
中文总结 AI 辅助
该研究针对图的直径等问题的敏感性设置，解决了递减场景的近似问题并改进了已有结果，同时提出递增场景的下界与匹配下界的新近似算法。
中文摘要 AI 辅助
敏感性设置是动态算法的一种受限设置，尤其适用于可进行大量预处理，但在数据结构最终重建前需对实时修改做出近乎即时响应的场景。对于图问题，会构造一个预处理时间为P的敏感性数据结构，使得以下查询能快速（最好在O(1)时间内）得到答案：给定一条边e，返回图G删除e（递减场景）或添加e（递增场景）后的问题答案。本文中，我们通过将预处理时间P匹配静态运行时间，并支持O(1)时间查询，几乎完全解决了多种近似场景下直径和离心率问题的递减场景，改进了此前针对单一边失效的所有结果[Bilò, Cohen, Friedrich, Schirneck, MFCS 2021；Bilò, Choudhary, Cohen, Friedrich, Krogmann, Schirneck, ICALP 2021]。更具体地：(1) 我们提供了一个紧归约，证明任何精确距离敏感性预言机都可用于高效解决递减场景下的精确直径和全节点离心率问题；(2) 对于近似场景，我们在所有稀疏性设置下，匹配了所有已知静态直径算法的运行时间，仅在近似度上额外有1+o(1)的因子。对于此前未探索的这些问题的递增场景：(3) 我们提出了新的下界，证明无向图中，任何递增算法都无法高效近似直径、半径或离心率，近似度超过5/3倍；有向图中则无法超过2倍；(4) 我们引入了两种新的启发性技术，并展示如何利用它们构造若干新算法。最值得注意的是，我们开发了有向图和无向图的递增单节点离心率近似算法，其性能匹配我们提出的新下界。
英文摘要
The sensitivity setting is a restricted setting for dynamic algorithms, particularly practical for scenarios where extensive preprocessing is feasible but responses to real-time modifications must be near-instantaneous before the data structure is eventually rebuilt. For graph problems, a sensitivity data structure is constructed with a preprocessing time P so that the following queries can be answered quickly, preferably in $O(1)$ time: given an edge $e$, return the answer to the problem on either $G \setminus e$ (decremental) or $G \cup e$ (incremental). In this paper, we almost entirely settle the decremental setting for the diameter and eccentricities problems in a variety of approximation regimes by matching P to the static runtime while supporting $O(1)$-time queries, thereby improving upon all previous results for a single failure [Bilò, Cohen, Friedrich, Schirneck, MFCS 2021; Bilò, Choudhary, Cohen, Friedrich, Krogmann, Schirneck, ICALP 2021]. More precisely: (1) We provide a tight reduction demonstrating that any exact distance sensitivity oracle can be used to efficiently solve decremental exact diameter and all-node eccentricities; (2) For the approximate setting, we match the runtime of all known static diameter algorithms across all sparsity settings, up to an additional $1+o(1)$ factor in approximation. Conversely, for the previously unexplored incremental setting of these problems: (3) We develop new lower bounds, demonstrating that no incremental algorithm can efficiently approximate diameter, radius, or eccentricity beyond a $5/3$ factor in undirected graphs or a $2$ factor in directed graphs; (4) We introduce two new instructive techniques and demonstrate how to utilize them to construct several new algorithms. Most notably, we develop incremental single-node eccentricity approximations for both directed and undirected graphs that match our new lower bounds.
