3 Search Results for "Magen, Ofer"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.57
Undirected 3-Fault Replacement Path in Nearly Cubic Time

Authors: Shucheng Chi, Ran Duan, Benyu Wang, and Tianle Xie

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
Given a graph G = (V,E) (n = |V|, m = |E|) and two vertices s,t ∈ V, the f-fault replacement path (fFRP) problem computes for every set F of at most f edges, the distance from s to t when edges in F fail. A recent result shows that 2FRP in directed graphs can be solved in Õ(n³) time [Vassilevska Williams, Woldeghebriel, Xu 2022]. In this paper, we show a 3FRP algorithm in deterministic Õ(n³) time for undirected weighted graphs, which almost matches the size of the output. This implies that fFRP in undirected graphs can be solved in nearly optimal Õ(n^f) time for all f ≥ 3. To construct our 3FRP algorithm, we introduce an incremental distance sensitivity oracle (DSO) for undirected graphs with Õ(n²) worst-case update time, while preprocessing time, space, and query time are still Õ(n³), Õ(n²) and Õ(1), respectively, which match the static DSO [Bernstein and Karger 2009]. Here in a DSO, we can preprocess a graph so that the distance between any pair of vertices given any failed edge can be answered efficiently. From the recent result in [Peng and Rubinstein 2023], we can obtain an offline dynamic DSO from the incremental worst-case DSO, which makes the construction of our 3FRP algorithm more convenient. By the offline dynamic DSO, we can also construct a 2-fault single-source replacement path (2-fault SSRP) algorithm in Õ(n³) time, that is, from a given vertex s, we want to find the distance to any vertex t when any pair of edges fail. Thus the Õ(n³) time complexity for 2-fault SSRP is also nearly optimal. Now we know that in undirected graphs 1FRP can be solved in Õ(m) time [Nardelli, Proietti, Widmayer 2001], and 2FRP and 3FRP in undirected graphs can be solved in Õ(n³) time. In this paper, we also show that a truly subcubic algorithm for 2FRP in undirected weighted graphs does not exist under APSP hypothesis.
Cite as

Shucheng Chi, Ran Duan, Benyu Wang, and Tianle Xie. Undirected 3-Fault Replacement Path in Nearly Cubic Time. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 57:1-57:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chi_et_al:LIPIcs.ICALP.2025.57,
  author =	{Chi, Shucheng and Duan, Ran and Wang, Benyu and Xie, Tianle},
  title =	{{Undirected 3-Fault Replacement Path in Nearly Cubic Time}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{57:1--57:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.57},
  URN =		{urn:nbn:de:0030-drops-234346},
  doi =		{10.4230/LIPIcs.ICALP.2025.57},
  annote =	{Keywords: Graph Algorithm, Shortest Path, Replacement Path}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.49
On Sparse Covers of Minor Free Graphs, Low Dimensional Metric Embeddings, and Other Applications

Authors: Arnold Filtser

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Given a metric space (X,d_X), a (β,s,Δ)-sparse cover is a collection of clusters 𝒞 ⊆ P(X) with diameter at most Δ, such that for every point x ∈ X, the ball B_X(x,Δ/β) is fully contained in some cluster C ∈ 𝒞, and x belongs to at most s clusters in 𝒞. Our main contribution is to show that the shortest path metric of every K_r-minor free graphs admits (O(r),O(r²),Δ)-sparse cover, and for every ε > 0, (4+ε,O(1/ε)^r,Δ)-sparse cover (for arbitrary Δ > 0). We then use this sparse cover to show that every K_r-minor free graph embeds into 𝓁_∞^{Õ(1/ε)^{r+1}⋅log n} with distortion 3+ε (resp. into 𝓁_∞^{Õ(r²)⋅log n} with distortion O(r)). Further, among other applications, this sparse cover immediately implies an algorithm for the oblivious buy-at-bulk problem in fixed minor free graphs with the tight approximation factor O(log n) (previously nothing beyond general graphs was known).
Cite as

Arnold Filtser. On Sparse Covers of Minor Free Graphs, Low Dimensional Metric Embeddings, and Other Applications. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 49:1-49:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{filtser:LIPIcs.SoCG.2025.49,
  author =	{Filtser, Arnold},
  title =	{{On Sparse Covers of Minor Free Graphs, Low Dimensional Metric Embeddings, and Other Applications}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{49:1--49:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.49},
  URN =		{urn:nbn:de:0030-drops-232015},
  doi =		{10.4230/LIPIcs.SoCG.2025.49},
  annote =	{Keywords: Sparse cover, minor free graphs, metric embeddings, 𝓁\underline∞, oblivious buy-at-bulk}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.81
Near Optimal Algorithm for the Directed Single Source Replacement Paths Problem

Authors: Shiri Chechik and Ofer Magen

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
In the Single Source Replacement Paths (SSRP) problem we are given a graph G = (V, E), and a shortest paths tree K̂ rooted at a node s, and the goal is to output for every node t ∈ V and for every edge e in K̂ the length of the shortest path from s to t avoiding e. We present an Õ(m√n + n²) time randomized combinatorial algorithm for unweighted directed graphs. Previously such a bound was known in the directed case only for the seemingly easier problem of replacement path where both the source and the target nodes are fixed. Our new upper bound for this problem matches the existing conditional combinatorial lower bounds. Hence, (assuming these conditional lower bounds) our result is essentially optimal and completes the picture of the SSRP problem in the combinatorial setting. Our algorithm naturally extends to the case of small, rational edge weights. In the full version of the paper, we strengthen the existing conditional lower bounds in this case by showing that any O(mn^(1/2-ε)) time (combinatorial or algebraic) algorithm for some fixed ε > 0 yields a truly sub-cubic algorithm for the weighted All Pairs Shortest Paths problem (previously such a bound was known only for the combinatorial setting).
Cite as

Shiri Chechik and Ofer Magen. Near Optimal Algorithm for the Directed Single Source Replacement Paths Problem. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 81:1-81:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chechik_et_al:LIPIcs.ICALP.2020.81,
  author =	{Chechik, Shiri and Magen, Ofer},
  title =	{{Near Optimal Algorithm for the Directed Single Source Replacement Paths Problem}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{81:1--81:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.81},
  URN =		{urn:nbn:de:0030-drops-124886},
  doi =		{10.4230/LIPIcs.ICALP.2020.81},
  annote =	{Keywords: Fault tolerance, Replacement Paths, Combinatorial algorithms, Conditional lower bounds}
}
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