AI 中文总结
针对部分动态无向加权图，提出基于稳定谱稀疏化的随机数据结构，可近优维护全对最大流与有效电阻的(1±ε)近似，在自适应对抗下高概率成功。
AI 中文摘要
我们提出了一种针对无向加权图的随机化数据结构，该图为部分动态图，即仅经历边插入或仅经历边删除。该数据结构可维护任意查询顶点对之间最大流值与有效电阻的(1±ε)近似值，总更新时间为$\tilde{O}_ε(n^2)$，最坏情况下查询时间为$\tilde{O}_ε(1)$。因此，对于$m = Ω(n^2)$的稠密图，我们的性能保证接近最优。我们的算法在自适应对抗环境下以高概率成功。\n我们的结果源于部分动态图的一个简单稳定性原理。我们展示了如何将包含$m$次更新的在线序列划分为$\tilde{O}(n/ε)$个时期，使得每个时期内的所有图都是该时期初始图的(1±O(ε))谱近似图。这些时期由更新边的累积杠杆分数决定：较小的杠杆分数质量意味着较小的谱变化，而单调更新序列上的总杠杆分数质量为$\tilde{O}(n)$。因此，谱稀疏化器每个时期仅需重新计算一次。将已知的静态全对最大流和有效电阻预言机应用于这些稀疏化器，即可得到上述结果。
英文摘要
We give a randomized data structure for undirected weighted graphs that are partially dynamic, i.e., that undergo either only edge insertions or only edge deletions. The data structure maintains $(1\pmε)$-approximations to the maxflow value and effective resistance between any queried pair of vertices, with total update time $\widetilde{O}_ε(n^2)$ and worst-case query time $\widetilde{O}_ε(1)$. Thus, for dense graphs where $m = Ω(n^2)$, our guarantees are near-optimal. Our algorithms succeed with high probability against an adaptive adversary. Our result follows from a simple stability principle for partially dynamic graphs. We show how to partition an online sequence of $m$ updates into $\widetilde{O}(n/ε)$ epochs such that every graph within an epoch is a $(1\pm O(ε))$-spectral approximation of the graph at the beginning of the epoch. The epochs are determined by the cumulative leverage score of the updated edges: small leverage-score mass implies small spectral change, while the total leverage-score mass over a monotone update sequence is $\widetilde{O}(n)$. Consequently, a spectral sparsifier needs to be recomputed only once per epoch. Applying known static all-pairs maxflow and effective-resistance oracles to these sparsifiers then yields the result.
