Dynamic Metric Embedding into $\ell_p$ Space

TL;DR

This paper presents the first dynamic algorithm for embedding weighted graphs into _p space with low distortion, efficiently handling edge weight increases, and establishes limitations in the fully dynamic setting.

Contribution

It introduces the first non-trivial decremental dynamic embedding algorithm into _p space with provable guarantees, extending Bourgain's static embedding result to dynamic graphs.

Findings

Expected distortion of the embedding is O( log^3 n).

Total update time is nearly linear in the number of edges and updates.

No low-distortion embedding can be maintained in fully dynamic regimes with high probability.

Abstract

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 \emph{distortion}, of their distance in $G$. Our main result is a dynamic algorithm with expected distortion $O(\log^3 n)$ and total update time $O\left((m^{1+o(1)} \log^2 W + Q \log n)\log(nW) \right)$, 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 growing 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.