Dynamic Metric Embedding into $\ell_p$ Space

TL;DR

The paper studies maintaining a low-distortion embedding of a dynamic graph metric into $\ell_p$ space, focusing on decremental updates where edge weights only increase. It develops a static Bourgain-type embedding built from distance-preserving cuts and then dynamically maintains these cuts via a decremental clustering/Bartal weak decomposition framework, achieving a polylogarithmic distortion $O(\log^3 n)$ with a dimension of $O(\log(nW))$ and a near-optimal update time. A key contribution is showing, unlike the decremental setting, that in the fully dynamic regime explicit maintenance of a low-distortion embedding is impossible w.h.p. with sublinear update time. This results in a practical, update-efficient dynamic embedding into $\ell_p$ that supports distance queries and has potential applications to dynamic graph problems and approximate all-pairs shortest paths in large-scale systems.

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 $φ: (G,d) \to (X,\ell_p)$ 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 $φ(u)$ and $φ(v)$ 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.

Figures (3)

• Figure 1: Adversarial sequence of graph updates
• Figure 2: Visualization of average distances in a dynamically changing metric. The orange line represents the average distance between all pairs of nodes computed exactly, using a deterministic shortest path algorithm, after every query. The blue line represents the average distance computed based on the embedding given by the dynamic embedding algorithm proposed in the paper.
• Figure 3: The ratio of the average distance between all pairs of points computed by of our embedding to the exact average distance between all pairs, after each query.

Theorems & Definitions (27)

• Theorem 1.1: bourgain1985lipschitz
• Theorem 1.2
• Theorem 1.3
• Definition 2.1: Maintain
• Theorem 3.1
• proof
• Definition 4.1: Distance preserving cut
• Lemma 4.2
• Definition 4.3
• Definition 4.4