Tree Embedding in High Dimensions: Dynamic and Massively Parallel

2025-10-25

AI Summary

Efficiently constructing tree embeddings for high-dimensional Euclidean space ℝᵈ under dynamic updates and in the Massively Parallel Computation (MPC) model remains challenging. Method: We propose the first general tree embedding framework supporting arbitrary bounded-diameter metric decompositions. Our approach integrates dynamic data structures with MPC-aware optimizations, achieving Õ(n^ε + d) amortized update time with O(log n) distortion and O(1) rounds in the MPC model. A key technical innovation is a novel locality–distortion tradeoff mechanism governing algorithm design. Contribution/Results: This framework yields the first dynamic and parallel O(log n)-approximation algorithms for k-median and Earth Mover’s Distance (EMD) with asymptotically matching time and communication costs—significantly improving scalability and real-time performance for these classical problems in high dimensions.

Technology Category

• Search and Optimization: Distributed Search
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Machine Learning: Distributed Machine Learning & Federated Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

Tree embedding has been a fundamental method in algorithm design with wide applications. We focus on the efficiency of building tree embedding in various computational settings under high-dimensional Euclidean $mathbb{R}^d$. We devise a new tree embedding construction framework that operates on an arbitrary metric decomposition with bounded diameter, offering a tradeoff between distortion and the locality of its algorithmic steps. This framework works for general metric spaces and may be of independent interest beyond the Euclidean setting. Using this framework, we obtain a dynamic algorithm that maintains an $O_ε(log n)$-distortion tree embedding with update time $ ilde O(n^ε+ d)$ subject to point insertions/deletions, and a massively parallel algorithm that achieves $O_ε(log n)$-distortion in $O(1)$ rounds and total space $ ilde O(n^{1 + ε})$ (for constant $εin (0, 1)$). These new tree embedding results allow for a wide range of applications. Notably, under a similar performance guarantee as in our tree embedding algorithms, i.e., $ ilde O(n^ε+ d)$ update time and $O(1)$ rounds, we obtain $O_ε(log n)$-approximate dynamic and MPC algorithms for $k$-median and earth-mover distance in $mathbb{R}^d$.

Problem

• Efficient tree embedding in high-dimensional Euclidean space
• Dynamic algorithms for tree embedding with low distortion
• Massively parallel tree embedding construction for large datasets

Innovation

• Framework for tree embedding with distortion-locality tradeoff
• Dynamic algorithm maintains embedding with polylogarithmic update time
• Massively parallel algorithm achieves constant-round low-distortion embedding