We give the first non-trivial decremental dynamic embedding of a weighted, undirected graph $G$ into $\ell_p$ space. Given a weighted graph $G$ undergoing a sequence of edge weight increases, the goal of this problem is to maintain a (randomized) mapping $\phi: (G,d) \to (X,\ell_p)$ from the set of vertices of the graph to the $\ell_p$ space such that for every pair of vertices $u$ and $v$, the expected distance between $\phi(u)$ and $\phi(v)$ in the $\ell_p$ metric is within a small multiplicative factor, referred to as the distortion, of their distance in $G$. Our main result is a dynamic algorithm with expected distortion $O(\log^2 n)$ and total update time $O\left((m^{1+o(1)} \log^2 W + Q)\log(nW) \right)$, where $W$ is the maximum weight of the edges, $Q$ is the total number of updates and $n, m$ denote the number of vertices and edges in $G$ respectively. This is the first result of its kind, extending the seminal result of Bourgain to the expanding field of dynamic algorithms. Moreover, we demonstrate that in the fully dynamic regime, where we tolerate edge insertions as well as deletions, no algorithm can explicitly maintain an embedding into $\ell_p$ space that has a low distortion with high probability.

翻译：我们首次给出了带权无向图 $G$ 到 $\ell_p$ 空间的非平凡递减动态嵌入。给定一个带权图 $G$ 经历一系列边权增加的操作，该问题的目标是维护一个从图顶点集到 $\ell_p$ 空间的（随机化）映射 $\phi: (G,d) \to (X,\ell_p)$，使得对于任意顶点对 $u$ 和 $v$，$\phi(u)$ 与 $\phi(v)$ 在 $\ell_p$ 度量下的期望距离，与其在图 $G$ 中的距离之比（称为失真度）在一个较小的乘法因子内。我们的主要成果是一个动态算法，其期望失真度为 $O(\log^2 n)$，总更新时间为 $O\left((m^{1+o(1)} \log^2 W + Q)\log(nW) \right)$，其中 $W$ 是边的最大权重，$Q$ 是总更新次数，$n$ 和 $m$ 分别表示 $G$ 的顶点数和边数。这是该领域的首个结果，将 Bourgain 的奠基性成果扩展到了动态算法这一新兴领域。此外，我们证明在完全动态场景下（即允许边插入和删除），任何算法都无法以高概率显式地维护一个具有低失真度的 $\ell_p$ 空间嵌入。