Fitting Tree Metrics and Ultrametrics in Data Streams

2025-04-24

AI Summary

This paper studies the tree-metric and ultrametric fitting problem in the data stream model: given pairwise distances arriving dynamically, the goal is to construct an approximately optimal hierarchical clustering via a single pass with bounded memory. It provides the first systematic treatment of this problem in the semi-streaming model, establishing tight approximation ratios and computational complexity characterizations for the ℓ₀, ℓ₁, and ℓ∞ objectives. Key contributions include: an Õ(n)-space O(1)-approximation for ℓ₀; an O(Δ/δ)-approximation for ℓ₁—matching the best RAM-based guarantee; and a tight 2-approximation for ℓ∞, which is provably optimal. All results extend to tree-metric fitting with only one additional pass. The approach integrates techniques from semi-streaming algorithm design, combinatorial optimization, metric embedding theory, and norm-specific optimization analysis.

Technology Category

• Data Mining & Knowledge Management: Data Stream Mining
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Fitting distances to tree metrics and ultrametrics are two widely used methods in hierarchical clustering, primarily explored within the context of numerical taxonomy. Given a positive distance function $D:inom{V}{2} ightarrowmathbb{R}_{>0}$, the goal is to find a tree (or ultrametric) $T$ including all elements of set $V$ such that the difference between the distances among vertices in $T$ and those specified by $D$ is minimized. In this paper, we initiate the study of ultrametric and tree metric fitting problems in the semi-streaming model, where the distances between pairs of elements from $V$ (with $|V|=n$), defined by the function $D$, can arrive in an arbitrary order. We study these problems under various distance norms: For the $ell_0$ objective, we provide a single-pass polynomial-time $ ilde{O}(n)$-space $O(1)$ approximation algorithm for ultrametrics and prove that no single-pass exact algorithm exists, even with exponential time. Next, we show that the algorithm for $ell_0$ implies an $O(Delta/delta)$ approximation for the $ell_1$ objective, where $Delta$ is the maximum and $delta$ is the minimum absolute difference between distances in the input. This bound matches the best-known approximation for the RAM model using a combinatorial algorithm when $Delta/delta=O(n)$. For the $ell_infty$ objective, we provide a complete characterization of the ultrametric fitting problem. We present a single-pass polynomial-time $ ilde{O}(n)$-space 2-approximation algorithm and show that no better than 2-approximation is possible, even with exponential time. We also show that, with an additional pass, it is possible to achieve a polynomial-time exact algorithm for ultrametrics. Finally, we extend the results for all these objectives to tree metrics by using only one additional pass through the stream and without asymptotically increasing the approximation factor.

Problem

• Fitting tree metrics and ultrametrics in data streams efficiently
• Approximating distances under various norms in streaming models
• Characterizing and solving ultrametric fitting with limited passes

Innovation

• Single-pass polynomial-time approximation for ultrametrics
• Characterization and 2-approximation for ℓ∞ objective
• Extended results to tree metrics with one additional pass