A geometric graph associated with a set of points $P= \{x_1, x_2, \cdots, x_n \} \subset \mathbb{R}^d$ and a fixed kernel function $\mathsf{K}:\mathbb{R}^d\times \mathbb{R}^d\to\mathbb{R}_{\geq 0}$ is a complete graph on $P$ such that the weight of edge $(x_i, x_j)$ is $\mathsf{K}(x_i, x_j)$. We present a fully-dynamic data structure that maintains a spectral sparsifier of a geometric graph under updates that change the locations of points in $P$ one at a time. The update time of our data structure is $n^{o(1)}$ with high probability, and the initialization time is $n^{1+o(1)}$. Under certain assumption, we can provide a fully dynamic spectral sparsifier with the robostness to adaptive adversary. We further show that, for the Laplacian matrices of these geometric graphs, it is possible to maintain random sketches for the results of matrix vector multiplication and inverse-matrix vector multiplication in $n^{o(1)}$ time, under updates that change the locations of points in $P$ or change the query vector by a sparse difference.
翻译:一组点的几何图形 $P= $x_ 1, x_ 2,\ cdots, x_n\ subset\ mathb{R ⁇ d$ 和固定内核函数 $\ mathsf{K} :\ mathbb{R ⁇ d\d\time\ mathbb{R ⁇ d\\to\mathb{R ⁇ d\\\\mathb{B{R ⁇ Geq0} 是一套美元美元的完整图表, 使( x_i, x_j) 的边缘重量为$mathsf{K} (x_i, x_j) $。 我们展示一个完全动态的数据结构, 在更新时以美元改变点的位置时, 我们的数据结构的更新时间是$n ⁇ 1+o(1)}, 初始化时间是$n ⁇ 1+o(1)} 美元。 在某种假设下, 我们可以提供一个完全动态的光谱的 spacespricspricricricricrence 和 $ $ rootostroostalticaltistrateal rodudeal rodudal rodulateal rodu the missal maxlusluslus rodulusals max max