2 Search Results for "Shabat, Idan"


Document
On the Size Overhead of Pairwise Spanners

Authors: Ofer Neiman and Idan Shabat

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)


Abstract
Given an undirected possibly weighted n-vertex graph G = (V,E) and a set 𝒫 ⊆ V² of pairs, a subgraph S = (V,E') is called a P-pairwise α-spanner of G, if for every pair (u,v) ∈ 𝒫 we have d_S(u,v) ≤ α⋅ d_G(u,v). The parameter α is called the stretch of the spanner, and its size overhead is define as |E'|/|P|. A surprising connection was recently discussed between the additive stretch of (1+ε,β)-spanners, to the hopbound of (1+ε,β)-hopsets. A long sequence of works showed that if the spanner/hopset has size ≈ n^{1+1/k} for some parameter k ≥ 1, then β≈(1/ε)^{log k}. In this paper we establish a new connection to the size overhead of pairwise spanners. In particular, we show that if |P|≈ n^{1+1/k}, then a P-pairwise (1+ε)-spanner must have size at least β⋅ |P| with β≈(1/ε)^{log k} (a near matching upper bound was recently shown in [Michael Elkin and Idan Shabat, 2023]). That is, the size overhead of pairwise spanners has similar bounds to the hopbound of hopsets, and to the additive stretch of spanners. We also extend the connection between pairwise spanners and hopsets to the large stretch regime, by showing nearly matching upper and lower bounds for P-pairwise α-spanners. In particular, we show that if |P|≈ n^{1+1/k}, then the size overhead is β≈k/α. A source-wise spanner is a special type of pairwise spanner, for which P = A×V for some A ⊆ V. A prioritized spanner is given also a ranking of the vertices V = (v₁,… ,v_n), and is required to provide improved stretch for pairs containing higher ranked vertices. By using a sequence of reductions: from pairwise spanners to source-wise spanners to prioritized spanners, we improve on the state-of-the-art results for source-wise and prioritized spanners. Since our spanners can be equipped with a path-reporting mechanism, we also substantially improve the known bounds for path-reporting prioritized distance oracles. Specifically, we provide a path-reporting distance oracle, with size O(n⋅(log log n)²), that has a constant stretch for any query that contains a vertex ranked among the first n^{1-δ} vertices (for any constant δ > 0). Such a result was known before only for non-path-reporting distance oracles.

Cite as

Ofer Neiman and Idan Shabat. On the Size Overhead of Pairwise Spanners. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 83:1-83:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{neiman_et_al:LIPIcs.ITCS.2024.83,
  author =	{Neiman, Ofer and Shabat, Idan},
  title =	{{On the Size Overhead of Pairwise Spanners}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{83:1--83:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.83},
  URN =		{urn:nbn:de:0030-drops-196110},
  doi =		{10.4230/LIPIcs.ITCS.2024.83},
  annote =	{Keywords: Graph Algorithms, Shortest Paths, Spanners}
}
Document
A Unified Framework for Hopsets

Authors: Ofer Neiman and Idan Shabat

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)


Abstract
Given an undirected graph G = (V,E), an (α,β)-hopset is a graph H = (V,E'), so that adding its edges to G guarantees every pair has an α-approximate shortest path that has at most β edges (hops), that is, d_G(u,v) ≤ d_{G∪H}^(β)(u,v) ≤ α⋅ d_G(u,v). Given the usefulness of hopsets for fundamental algorithmic tasks, several different algorithms and techniques were developed for their construction, for various regimes of the stretch parameter α. In this work we devise a single algorithm that can attain all state-of-the-art hopsets for general graphs, by choosing the appropriate input parameters. In fact, in some cases it also improves upon the previous best results. We also show a lower bound on our algorithm. In [Ben-Levy and Parter, 2020], given a parameter k, a (O(k^ε),O(k^{1-ε}))-hopset of size Õ(n^{1+1/k}) was shown for any n-vertex graph and parameter 0 < ε < 1, and they asked whether this result is best possible. We resolve this open problem, showing that any (α,β)-hopset of size O(n^{1+1/k}) must have α⋅β ≥ Ω(k).

Cite as

Ofer Neiman and Idan Shabat. A Unified Framework for Hopsets. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 81:1-81:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


Copy BibTex To Clipboard

@InProceedings{neiman_et_al:LIPIcs.ESA.2022.81,
  author =	{Neiman, Ofer and Shabat, Idan},
  title =	{{A Unified Framework for Hopsets}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{81:1--81:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.81},
  URN =		{urn:nbn:de:0030-drops-170192},
  doi =		{10.4230/LIPIcs.ESA.2022.81},
  annote =	{Keywords: Graph Algorithms, Shortest Paths, Hopsets}
}
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