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Tree Embedding in High Dimensions: Dynamic and Massively Parallel

Gramoz Goranci, Shaofeng H. -C. Jiang, Peter Kiss, Qihao Kong, Yi Qian, Eva Szilagyi

TL;DR

This work introduces a general framework for tree embeddings built from bounded-diameter metric decompositions, enabling distortion of $O_\epsilon(\log n)$ while balancing locality via a parameter $\Gamma$. The authors develop both fully dynamic and MPC implementations in high-dimensional Euclidean spaces, leveraging sparse partitions and consistent hashing to achieve near-optimal distortion with sublinear to polylog update costs. The framework translates to powerful dynamic algorithms for core geometric problems, including $k$-median, Euclidean MST, EMD, and various matching/transport tasks, with provable guarantees in both dynamic and distributed (MPC) models. The results significantly advance the practicality of tree-embedding-based techniques in high dimensions, providing scalable methods for continuous updates and parallel computation. Overall, the paper delivers a unified, general-purpose embedding approach and demonstrates its broad applicability and efficiency in dynamic and MPC contexts, with implications for clustering, matching, and transport in large-scale, high-dimensional data analyses.

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 $\tilde O(n^ε+ d)$ subject to point insertions/deletions, and a massively parallel algorithm that achieves $O_ε(\log n)$-distortion in $O(1)$ rounds and total space $\tilde O(n^{1 + ε})$ (for constant $ε\in (0, 1)$). 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., $\tilde O(n^ε+ d)$ 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 $\mathbb{R}^d$.

Tree Embedding in High Dimensions: Dynamic and Massively Parallel

TL;DR

This work introduces a general framework for tree embeddings built from bounded-diameter metric decompositions, enabling distortion of while balancing locality via a parameter . The authors develop both fully dynamic and MPC implementations in high-dimensional Euclidean spaces, leveraging sparse partitions and consistent hashing to achieve near-optimal distortion with sublinear to polylog update costs. The framework translates to powerful dynamic algorithms for core geometric problems, including -median, Euclidean MST, EMD, and various matching/transport tasks, with provable guarantees in both dynamic and distributed (MPC) models. The results significantly advance the practicality of tree-embedding-based techniques in high dimensions, providing scalable methods for continuous updates and parallel computation. Overall, the paper delivers a unified, general-purpose embedding approach and demonstrates its broad applicability and efficiency in dynamic and MPC contexts, with implications for clustering, matching, and transport in large-scale, high-dimensional data analyses.

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 . 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 -distortion tree embedding with update time subject to point insertions/deletions, and a massively parallel algorithm that achieves -distortion in rounds and total space (for constant ). 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., update time and rounds, we obtain -approximate dynamic and MPC algorithms for -median and earth-mover distance in .
Paper Structure (78 sections, 33 theorems, 42 equations, 12 algorithms)

This paper contains 78 sections, 33 theorems, 42 equations, 12 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Dynamic Tree Embedding in $\mathbb{R}^d$.
  4. Applications of Dynamic Tree Embedding.
  5. MPC Tree Embeddings.
  6. Technical Contributions
  7. Tree Embedding via General Metric Decomposition.
  8. Efficient Implementations in Euclidean $\mathbb{R}^d$.
  9. Proof Overview: Tree Embedding via General Metric Decomposition
  10. Background: CKR Decomposition CalinescuKR04.
  11. Approximating CKR via Metric Decomposition.
  12. Interpreting Metric Decomposition as Hashing.
  13. Our Modified CKR Algorithm and Tree Embedding.
  14. Distortion Analysis: Review of DBLP:journals/ejcss/FakcharoenpholRT04 Analysis.
  15. Distortion Analysis: Technical Challenges.
  16. ...and 63 more sections

Key Result

Theorem 1.1

There is a fully dynamic algorithm such that for any $\epsilon \in (0,1)$, it supports insertions and deletions of points from Euclidean $\mathbb{R}^d$ and maintains a probabilistic tree embedding with $O_{\epsilon}(\log n)$ distortion in $\tilde{O}(n^{\epsilon})$ expected amortized update time.

Theorems & Definitions (66)

  • Theorem 1.1: Dynamic Tree Embedding; see \ref{['thm:dynamic']}
  • Corollary 1.2: Dynamic $k$-median Clustering; see \ref{['thm:dyn_kmedian']}
  • Corollary 1.3: Dynamic Euclidean Bipartite Matching; see \ref{['thm:EMD']}
  • Theorem 1.4: MPC Tree Embedding; see \ref{['thm:mpc']}
  • Remark
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.2
  • proof
  • ...and 56 more