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Streaming Algorithms for Geometric Steiner Forest

Authors Artur Czumaj, Shaofeng H.-C. Jiang , Robert Krauthgamer, Pavel Veselý

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Author Details

Artur Czumaj
  • University of Warwick, Coventry, UK
Shaofeng H.-C. Jiang
  • Peking University, Beijing, China
Robert Krauthgamer
  • Weizmann Institute of Science, Rehovot, Israel
Pavel Veselý
  • Charles University, Prague, Czech Republic

Funding

Work partially supported by the National key R & D program of China No. 2021YFA1000900.
  • Czumaj, Artur: Research partially supported by the Centre for Discrete Mathematics and its Applications (DIMAP), by a Weizmann-UK Making Connections Grant, by an IBM Faculty Award, and by EPSRC award EP/V01305X/1.
  • Jiang, Shaofeng H.-C.: Part of this work was done when the author was at the Weizmann Institute of Science and Aalto University. Partially suppported by a startup fund from Peking University, and the Advanced Institute of Information Technology, Peking University.
  • Krauthgamer, Robert: Work partially supported by ONR Award N00014-18-1-2364, the Israel Science Foundation grant #1086/18, a Weizmann-UK Making Connections Grant, the Weizmann Data Science Research Center, and a Minerva Foundation grant.
  • Veselý, Pavel: Part of this work was done when the author was at the University of Warwick. Partially supported by European Research Council grant ERC-2014-CoG 647557, by a Weizmann-UK Making Connections Grant, by GA ČR project 19-27871X, and by Charles University project UNCE/SCI/004.

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Artur Czumaj, Shaofeng H.-C. Jiang, Robert Krauthgamer, and Pavel Veselý. Streaming Algorithms for Geometric Steiner Forest. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 47:1-47:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.47

Abstract

We consider an important generalization of the Steiner tree problem, the Steiner forest problem, in the Euclidean plane: the input is a multiset X ⊆ ℝ², partitioned into k color classes C₁, C₂, …, Cₖ ⊆ X. The goal is to find a minimum-cost Euclidean graph G such that every color class Cᵢ is connected in G. We study this Steiner forest problem in the streaming setting, where the stream consists of insertions and deletions of points to X. Each input point x ∈ X arrives with its color color(x) ∈ [k], and as usual for dynamic geometric streams, the input is restricted to the discrete grid {0, …, Δ}².
We design a single-pass streaming algorithm that uses poly(k ⋅ log Δ) space and time, and estimates the cost of an optimal Steiner forest solution within ratio arbitrarily close to the famous Euclidean Steiner ratio α₂ (currently 1.1547 ≤ α₂ ≤ 1.214). This approximation guarantee matches the state of the art bound for streaming Steiner tree, i.e., when k = 1. Our approach relies on a novel combination of streaming techniques, like sampling and linear sketching, with the classical Arora-style dynamic-programming framework for geometric optimization problems, which usually requires large memory and has so far not been applied in the streaming setting.
We complement our streaming algorithm for the Steiner forest problem with simple arguments showing that any finite approximation requires Ω(k) bits of space.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sketching and sampling
  • Theory of computation → Streaming models
  • Theory of computation → Computational geometry
Keywords
  • Steiner forest
  • streaming
  • sublinear algorithms
  • dynamic programming

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