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 minimumcost 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 singlepass 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 Arorastyle dynamicprogramming 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.
BibTeX  Entry
@InProceedings{czumaj_et_al:LIPIcs.ICALP.2022.47,
author = {Czumaj, Artur and Jiang, Shaofeng H.C. and Krauthgamer, Robert and Vesel\'{y}, Pavel},
title = {{Streaming Algorithms for Geometric Steiner Forest}},
booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
pages = {47:147:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772358},
ISSN = {18688969},
year = {2022},
volume = {229},
editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16388},
URN = {urn:nbn:de:0030drops163880},
doi = {10.4230/LIPIcs.ICALP.2022.47},
annote = {Keywords: Steiner forest, streaming, sublinear algorithms, dynamic programming}
}
Keywords: 

Steiner forest, streaming, sublinear algorithms, dynamic programming 
Collection: 

49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) 
Issue Date: 

2022 
Date of publication: 

28.06.2022 