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Fitting Tree Metrics and Ultrametrics in Data Streams

Authors Amir Carmel , Debarati Das , Evangelos Kipouridis , Evangelos Pipis

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  • Theory of computation → Streaming models
  • Theory of computation → Sketching and sampling
  • Theory of computation → Facility location and clustering
Keywords
  • Streaming
  • Clustering
  • Ultrametrics
  • Tree metrics
  • Distance fitting

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Abstract

Fitting distances to tree metrics and ultrametrics are two widely used methods in hierarchical clustering, primarily explored within the context of numerical taxonomy. Formally, given a positive distance function D: binom(V,2) → ℝ_{>0}, the goal is to find a tree (or an 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. Numerical taxonomy was first introduced by Sneath and Sokal [Nature 1962], and since then it has been studied extensively in both biology and computer science.
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; namely the 𝓁₀ objective, which aims to minimize the number of modified entries in D to fit a tree-metric or an ultrametric; the 𝓁₁ objective, which seeks to minimize the total sum of distance errors across all pairs of points in V; and the 𝓁_∞ objective, which focuses on minimizing the maximum error incurred by any entries in D.
- Our first result addresses the 𝓁₀ objective. We provide a single-pass polynomial-time Õ(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 𝓁₀ implies an O(Δ/δ) approximation for the 𝓁₁ objective, where Δ is the maximum, and δ 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 Δ/δ = O(n).
- For the 𝓁_∞ objective, we provide a complete characterization of the ultrametric fitting problem. First, we present a single-pass polynomial-time Õ(n)-space 2-approximation algorithm and show that no better than 2-approximation is possible, even with exponential time. Furthermore, we show that with an additional pass, it is possible to achieve a polynomial-time exact algorithm for ultrametrics.
- Finally, we extend all these results to tree metrics by using only one additional pass through the stream and without asymptotically increasing the approximation factor.

Cite As Get BibTex

Amir Carmel, Debarati Das, Evangelos Kipouridis, and Evangelos Pipis. Fitting Tree Metrics and Ultrametrics in Data Streams. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 42:1-42:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.42

Author Details

Amir Carmel
  • Pennsylvania State University, State College, PA, USA
Debarati Das
  • Pennsylvania State University, State College, PA, USA
Evangelos Kipouridis
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Evangelos Pipis
  • National Technical University of Athens, Greece
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany

Funding

  • Das, Debarati: Research supported by NSF grant 2337832.

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