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DOI: 10.4230/LIPIcs.FSTTCS.2024.29

Matchings in Low-Arboricity Graphs in the Dynamic Graph Stream Model

Authors: Christian Konrad, Andrew McGregor, Rik Sengupta, and Cuong Than

Published in: LIPIcs, Volume 323, 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)

Abstract

We consider the problem of estimating the size of a maximum matching in low-arboricity graphs in the dynamic graph stream model. In this setting, an algorithm with limited memory makes multiple passes over a stream of edge insertions and deletions, resulting in a low-arboricity graph. Let n be the number of vertices of the input graph, and α be its arboricity. We give the following results. 1) As our main result, we give a three-pass streaming algorithm that produces an (α + 2)(1 + ε)-approximation and uses space O(ε^{-2}⋅α²⋅n^{1/2}⋅log n). This result should be contrasted with the Ω(α^{-5/2}⋅n^{1/2}) space lower bound established by [Assadi et al., SODA'17] for one-pass algorithms, showing that, for graphs of constant arboricity, the one-pass space lower bound can be achieved in three passes (up to poly-logarithmic factors). Furthermore, we obtain a two-pass algorithm that uses space O(ε^{-2}⋅α²⋅n^{3/5}⋅log n). 2) We also give a (1+ε)-approximation multi-pass algorithm, where the space used is parameterized by an upper bound on the size of a largest matching. For example, using O(log log n) passes, the space required is O(ε^{-1}⋅α²⋅k⋅log n), where k denotes an upper bound on the size of a largest matching. Finally, we define a notion of arboricity in the context of matrices. This is a natural measure of the sparsity of a matrix that is more nuanced than simply bounding the total number of nonzero entries, but less restrictive than bounding the number of nonzero entries in each row and column. For such matrices, we exploit our results on estimating matching size to present upper bounds for the problem of rank estimation in the dynamic data stream model.

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Christian Konrad, Andrew McGregor, Rik Sengupta, and Cuong Than. Matchings in Low-Arboricity Graphs in the Dynamic Graph Stream Model. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{konrad_et_al:LIPIcs.FSTTCS.2024.29,
  author =	{Konrad, Christian and McGregor, Andrew and Sengupta, Rik and Than, Cuong},
  title =	{{Matchings in Low-Arboricity Graphs in the Dynamic Graph Stream Model}},
  booktitle =	{44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)},
  pages =	{29:1--29:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-355-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{323},
  editor =	{Barman, Siddharth and Lasota, S{\l}awomir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2024.29},
  URN =		{urn:nbn:de:0030-drops-222187},
  doi =		{10.4230/LIPIcs.FSTTCS.2024.29},
  annote =	{Keywords: Data Streams, Graph Matching, Graph Arboricity}
}

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DOI: 10.4230/LIPIcs.SoCG.2024.37

Optimal Euclidean Tree Covers

Authors: Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenković, Shay Solomon, and Cuong Than

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract

A (1+e)-stretch tree cover of a metric space is a collection of trees, where every pair of points has a (1+e)-stretch path in one of the trees. The celebrated Dumbbell Theorem [Arya et al. STOC'95] states that any set of n points in d-dimensional Euclidean space admits a (1+e)-stretch tree cover with O_d(e^{-d} ⋅ log(1/e)) trees, where the O_d notation suppresses terms that depend solely on the dimension d. The running time of their construction is O_d(n log n ⋅ log(1/e)/e^d + n ⋅ e^{-2d}). Since the same point may occur in multiple levels of the tree, the maximum degree of a point in the tree cover may be as large as Ω(log Φ), where Φ is the aspect ratio of the input point set. In this work we present a (1+e)-stretch tree cover with O_d(e^{-d+1} ⋅ log(1/e)) trees, which is optimal (up to the log(1/e) factor). Moreover, the maximum degree of points in any tree is an absolute constant for any d. As a direct corollary, we obtain an optimal {routing scheme} in low-dimensional Euclidean spaces. We also present a (1+e)-stretch Steiner tree cover (that may use Steiner points) with O_d(e^{(-d+1)/2} ⋅ log(1/e)) trees, which too is optimal. The running time of our two constructions is linear in the number of edges in the respective tree covers, ignoring an additive O_d(n log n) term; this improves over the running time underlying the Dumbbell Theorem.

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Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenković, Shay Solomon, and Cuong Than. Optimal Euclidean Tree Covers. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chang_et_al:LIPIcs.SoCG.2024.37,
  author =	{Chang, Hsien-Chih and Conroy, Jonathan and Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay and Than, Cuong},
  title =	{{Optimal Euclidean Tree Covers}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{37:1--37:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.37},
  URN =		{urn:nbn:de:0030-drops-199828},
  doi =		{10.4230/LIPIcs.SoCG.2024.37},
  annote =	{Keywords: Tree cover, spanner, Steiner point, routing, bounded-degree, quadtree, net-tree}
}

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Matchings in Low-Arboricity Graphs in the Dynamic Graph Stream Model

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Optimal Euclidean Tree Covers

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