17 Search Results for "Cohen, Sarel"

Document
DOI: 10.4230/LIPIcs.FSTTCS.2025.27
Fault-Tolerant Approximate Distance Oracles with a Source Set

Authors: Dipan Dey and Telikepalli Kavitha

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract
Our input is an undirected weighted graph G = (V,E) on n vertices along with a source set S ⊆ V. The problem is to preprocess G and build a compact data structure such that upon query Qu(s,v,f) where (s,v) ∈ S×V and f is any faulty edge, we can quickly find a good estimate (i.e., within a small multiplicative stretch) of the s-v distance in G-f. We use a fault-tolerant ST-distance oracle from the work of Bilò et al. (STACS 2018) to construct an S×V approximate distance oracle or sourcewise approximate distance oracle of size Õ(|S|n + n^{3/2}) with multiplicative stretch at most 5. We construct another fault-tolerant sourcewise approximate distance oracle of size Õ(|S|n + n^{4/3}) with multiplicative stretch at most 13. Both the oracles have O(1) query answering time.
Cite as

Dipan Dey and Telikepalli Kavitha. Fault-Tolerant Approximate Distance Oracles with a Source Set. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dey_et_al:LIPIcs.FSTTCS.2025.27,
  author =	{Dey, Dipan and Kavitha, Telikepalli},
  title =	{{Fault-Tolerant Approximate Distance Oracles with a Source Set}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{27:1--27:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.27},
  URN =		{urn:nbn:de:0030-drops-251081},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.27},
  annote =	{Keywords: Weighted graphs, approximate distances, fault-tolerant data structures}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2025.43
New Approximate Distance Oracles and Their Applications

Authors: Avi Kadria and Liam Roditty

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract
Let G = (V, E) be an undirected graph with n vertices and m edges, and let μ = m/n. A distance oracle is a data structure designed to answer approximate distance queries, with the goal of achieving low stretch, efficient space usage, and fast query time. While much of the prior work focused on distance oracles with constant query time, this paper presents a comprehensive study of distance oracles with non-constant query time. We explore the tradeoffs between space, stretch, and query time of distance oracles in various regimes. Specifically, we consider both weighted and unweighted graphs in the regimes of stretch < 2 and stretch ≥ 2. In addition, we demonstrate several applications of our new distance oracles to the n-Pairs Shortest Paths (n-PSP) problem and the All Nodes Shortest Cycles (ANSC) problem. Our main contributions are: - Weighted graphs: We present a new three-way trade-off between stretch, space, and query time, offering a natural extension of the classical Thorup–Zwick distance oracle [STOC’01 and JACM’05] to regimes with larger query time. Specifically, for any 0 < r < 1/2 and integer k ≥ 1, we construct a (2k(1 - 2r) - 1)-stretch distance oracle with Õ(m + n^{1 + 1/k}) space and Õ(μ n^r) query time. This construction provides an asymptotic improvement over the classical (2k - 1)-stretch and O(n^{1 + 1/k})-space tradeoff of Thorup and Zwick in sparse graphs, at the cost of increased query time. We also improve upon a result of Dalirrooyfard et al. [FOCS’22], who presented a (2k - 2)-stretch distance oracle with O(m + n^{1 + 1/k}) space and O(μ n^{1/k}) query time. In our oracle we reduce the stretch from (2k - 2) to (2k - 5) while preserving the same space and query time. - Unweighted graphs: We present a (2k - 5, 4 + 2_{odd})-approximation distance oracle with O(n^{1 + 1/k}) space and O(n^{1/k}) query time. This improves upon a (2k - 2, 2_{odd})-approximation distance oracle of Dalirrooyfard et al. [FOCS’22] while maintaining the same space and query time. We also present a distance oracle that given u,v ∈ V returns an estimate d̂(u,v) ≤ d(u,v) + 2⌈ d(u,v) / 3 ⌉ + 2, using O(n^{4/3 + 2ε}) space and O(n^{1 - 3ε}) query time. To the best of our knowledge, this is the first distance oracle that simultaneously achieves a multiplicative stretch < 2, and a space complexity O(n^{1.5 - α}), for some α > 0. - Applications for n-PSP and ANSC: We present an Õ(m^{1 - 1/(k+1)} n)-time algorithm for the n-PSP problem, that for every input pair ⟨s_i,t_i⟩, where i ∈ [n], returns an estimate d̂(s_i, t_i) such that d̂(s_i,t_i) ≤ d(s_i,t_i) + 2⌈d(s_i,t_i)/2k⌉. By allowing a small additive error, this result circumvents the conditional running time lower bound of Ω(m^{2 - 2/(k+1)} ⋅ n^{1/(k+1) - o(1)}), established by Dalirrooyfard et al. [FOCS’22] for achieving (1 + 1/k)-stretch. Additionally, we present an Õ(mn^{1 - 1/k})-time algorithm for the ANSC problem that computes, for every u ∈ V, an estimate ĉ_u such that ĉ_u ≤ SC(u) + 2⌈SC(u)/2(k - 1)⌉, where SC(u) denotes the length of the shortest cycle containing u. This improves upon the Õ(m^{2 - 2/k}n^{1/k})-time algorithm of Dalirrooyfard et al. [FOCS'22], while achieving the same approximation guarantee. We obtain our results by developing several new techniques, among them are the borderline vertices technique and the middle vertex technique, which may be of independent interest.
Cite as

Avi Kadria and Liam Roditty. New Approximate Distance Oracles and Their Applications. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 43:1-43:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kadria_et_al:LIPIcs.ISAAC.2025.43,
  author =	{Kadria, Avi and Roditty, Liam},
  title =	{{New Approximate Distance Oracles and Their Applications}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{43:1--43:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.43},
  URN =		{urn:nbn:de:0030-drops-249514},
  doi =		{10.4230/LIPIcs.ISAAC.2025.43},
  annote =	{Keywords: Distance oracles, Fine-grained algorithms, Graph algorithms, Data structures}
}
Document
DOI: 10.4230/LIPIcs.DISC.2025.40
On the Randomized Locality of Matching Problems in Regular Graphs

Authors: Seri Khoury, Manish Purohit, Aaron Schild, and Joshua R. Wang

Published in: LIPIcs, Volume 356, 39th International Symposium on Distributed Computing (DISC 2025)

Abstract
The main goal in distributed symmetry-breaking is to understand the locality of problems: the radius of the neighborhood that a node must explore to determine its part of a global solution. In this work, we study the locality of matching problems in the family of regular graphs, which is one of the main benchmarks for establishing lower bounds on the locality of symmetry-breaking problems, as well as for obtaining classification results. Our main results are summarized as follows: 1) Approximate matching: We develop randomized algorithms to show that (1 + ε)-approximate matching in regular graphs is truly local, i.e., the locality depends only on ε and is independent of all other graph parameters. Furthermore, as long as the degree Δ is not very small (namely, as long as Δ ≥ poly(1/ε)), this dependence is only logarithmic in 1/ε. This stands in sharp contrast to maximal matching in regular graphs which requires some dependence on the number of nodes n or the degree Δ. 2) Maximal matching: Our techniques further allow us to establish a strong separation between the node-averaged complexity and worst-case complexity of maximal matching in regular graphs, by showing that the former is only O(1). Central to our main technical contribution is a novel martingale-based analysis for the ≈ 40-year-old algorithm by Luby. In particular, our analysis shows that applying one round of Luby’s algorithm on the line graph of a Δ-regular graph results in an almost Δ/2-regular graph.
Cite as

Seri Khoury, Manish Purohit, Aaron Schild, and Joshua R. Wang. On the Randomized Locality of Matching Problems in Regular Graphs. In 39th International Symposium on Distributed Computing (DISC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 356, pp. 40:1-40:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{khoury_et_al:LIPIcs.DISC.2025.40,
  author =	{Khoury, Seri and Purohit, Manish and Schild, Aaron and Wang, Joshua R.},
  title =	{{On the Randomized Locality of Matching Problems in Regular Graphs}},
  booktitle =	{39th International Symposium on Distributed Computing (DISC 2025)},
  pages =	{40:1--40:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-402-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{356},
  editor =	{Kowalski, Dariusz R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2025.40},
  URN =		{urn:nbn:de:0030-drops-248570},
  doi =		{10.4230/LIPIcs.DISC.2025.40},
  annote =	{Keywords: regular graphs, maximum matching, augmenting paths, distributed algorithms, Luby’s algorithm, martingales}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.68
Faster Algorithm for Second (s,t)-Mincut and Breaking Quadratic Barrier for Dual Edge Sensitivity for (s,t)-Mincut

Authors: Surender Baswana, Koustav Bhanja, and Anupam Roy

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
Let G be a directed graph on n vertices and m edges. In this article, we study (s,t)-cuts of second minimum capacity and present the following algorithmic and graph-theoretic results. 1) Second (s,t)-mincut: Vazirani and Yannakakis [ICALP 1992] designed the first algorithm for computing an (s,t)-cut of second minimum capacity using {O}(n²) maximum (s,t)-flow computations. We present the following algorithm that improves the running time significantly. For directed integer-weighted graphs, there is an algorithm that can compute an (s,t)-cut of second minimum capacity using Õ(√n) maximum (s,t)-flow computations with high probability. To achieve this result, a close relationship of independent interest is established between (s,t)-cuts of second minimum capacity and global mincuts in directed weighted graphs. 2) Minimum+1 (s,t)-cuts: Minimum+1 (s,t)-cuts have been studied quite well recently [Baswana, Bhanja, and Pandey, ICALP 2022 & TALG 2023], which is a special case of second (s,t)-mincut. We present the following structural result and the first nontrivial algorithm for minimum+1 (s,t)-cuts. 3) Algorithm: For directed multi-graphs, we design an algorithm that, given any maximum (s,t)-flow, computes a minimum+1 (s,t)-cut, if it exists, in O(m) time. 4) Structure: The existing structures for storing and characterizing all minimum+1 (s,t)-cuts occupy {O}(mn) space [Baswana, Bhanja, and Pandey, TALG 2023]. For undirected multi-graphs, we design a directed acyclic graph (DAG) occupying only {O}(m) space that stores and characterizes all minimum+1 (s,t)-cuts. This matches the space bound of the widely-known DAG structure for all (s,t)-mincuts [Picard and Queyranne, Math. Prog. Studies 1980]. 5) Dual Edge Sensitivity Oracle: The study of minimum+1 (s,t)-cuts often turns out to be useful in designing dual edge sensitivity oracles - a compact data structure for efficiently reporting an (s,t)-mincut after insertion/failure of any given pair of query edges. It has been shown recently [Bhanja, ICALP 2025] that any dual edge sensitivity oracle for (s,t)-mincut in undirected multi-graphs must occupy Ω(n²) space in the worst-case irrespective of the query time. Interestingly, for undirected unweighted simple graphs, we break this quadratic barrier while achieving a non-trivial query time as follows. There is an O(n√n) space data structure that can report an (s,t)-mincut in O(min{m,n√n}) time after the insertion/failure of any given pair of query edges. To arrive at our results, as one of our key techniques, we establish interesting relationships between (s,t)-cuts of capacity (minimum+Δ), Δ ≥ 0, and maximum (s,t)-flow. We believe that these techniques and the graph-theoretic result in 2.(b) are of independent interest.
Cite as

Surender Baswana, Koustav Bhanja, and Anupam Roy. Faster Algorithm for Second (s,t)-Mincut and Breaking Quadratic Barrier for Dual Edge Sensitivity for (s,t)-Mincut. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 68:1-68:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{baswana_et_al:LIPIcs.ESA.2025.68,
  author =	{Baswana, Surender and Bhanja, Koustav and Roy, Anupam},
  title =	{{Faster Algorithm for Second (s,t)-Mincut and Breaking Quadratic Barrier for Dual Edge Sensitivity for (s,t)-Mincut}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{68:1--68:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.68},
  URN =		{urn:nbn:de:0030-drops-245369},
  doi =		{10.4230/LIPIcs.ESA.2025.68},
  annote =	{Keywords: mincut, second mincut, compact structure, fault tolerant, sensitivity oracle, dual edges, st mincut, global mincut, characterization}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.27
Minimum+1 Steiner Cut and Dual Edge Sensitivity Oracle: Bridging Gap between Global and (s,t)-cut

Authors: Koustav Bhanja

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

Abstract
Let G = (V,E) be an undirected multi-graph on n = |V| vertices and S ⊆ V be a Steiner set in G. Steiner cut is a fundamental concept; moreover, global cut (|S| = n), as well as (s,t)-cut (|S| = 2), is just a special case of Steiner cut. We study Steiner cuts of capacity minimum+1, and as an important application, we provide a dual edge Sensitivity Oracle for Steiner mincut - a compact data structure for efficiently reporting a Steiner mincut after failure/insertion of any pair of edges. A compact data structure for cuts of capacity minimum+1 has been designed for both global cuts [Dinitz and Nutov, STOC 1995] and (s,t)-cuts [Baswana, Bhanja, and Pandey, ICALP 2022 & TALG 2023]. Moreover, both data structures are also used crucially to design a dual edge Sensitivity Oracle for their respective mincuts. Unfortunately, except for these two extreme scenarios of Steiner cuts, no generalization of these results is known. Therefore, to address this gap, we present the following first results on Steiner cuts for any S satisfying 2 ≤ |S| ≤ n. 1) Data Structure for Minimum+1 Steiner Cut: There is an {O}(n(n-|S|+1)) space data structure that, given any pair of vertices u,v, can determine in {O}(1) time whether the Steiner cut of the least capacity separating u and v has capacity minimum+1. It can report such a cut, if it exists, in {O}(n) time, which is worst-case optimal. 2) Dual Edge Sensitivity Oracle: We design the following pair of data structures. (a) There is an {O}(n(n-|S|+1)) space data structure that, after the failure or insertion of any pair of edges in G, can report the capacity of Steiner mincut in {O}(1) time and a Steiner mincut in {O}(n) time, which is worst-case optimal. (b) If we are interested in reporting only the capacity of Steiner mincut, there is a more compact data structure that occupies {O}((n-|S|)²+n) space and can report the capacity of Steiner mincut in {O}(1) time after the failure or insertion of any pair of edges. 3) Lower Bound for Sensitivity Oracle: For undirected multi-graphs, for any Steiner set S ⊆ V, any data structure that, after the failure or insertion of any pair of edges, can report the capacity of Steiner mincut must occupy Ω((n-|S|)²) bits of space in the worst case, irrespective of the query time. To arrive at our results, we provide several techniques, especially a generalization of the 3-Star Lemma given by Dinitz and Vainshtein [SICOMP 2000], which is of independent interest. Our results achieve the same space and time bounds of the existing results for the two extreme scenarios of Steiner cuts - global and (s,t)-cut. In addition, the space occupied by our data structures in (1) and (2) reduces as |S| tends to n. Also, they occupy subquadratic space if |S| is close to n.
Cite as

Koustav Bhanja. Minimum+1 Steiner Cut and Dual Edge Sensitivity Oracle: Bridging Gap between Global and (s,t)-cut. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 27:1-27:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bhanja:LIPIcs.ICALP.2025.27,
  author =	{Bhanja, Koustav},
  title =	{{Minimum+1 Steiner Cut and Dual Edge Sensitivity Oracle: Bridging Gap between Global and (s,t)-cut}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{27:1--27:20},
  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.2025.27},
  URN =		{urn:nbn:de:0030-drops-234040},
  doi =		{10.4230/LIPIcs.ICALP.2025.27},
  annote =	{Keywords: cut, mincut, minimum+1, steiner, edge fault, sensitivity oracle, dual edges}
}
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.SAND.2025.6
Dismountability in Temporal Cliques Revisited

Authors: Daniele Carnevale, Arnaud Casteigts, and Timothée Corsini

Published in: LIPIcs, Volume 330, 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)

Abstract
A temporal graph is a graph whose edges are available only at certain points in time. It is temporally connected if the nodes can reach each other by paths that traverse the edges chronologically (temporal paths). Unlike static graphs, temporal graphs do not always admit small subsets of edges that preserve connectivity (temporal spanners) - there exist temporal graphs with Θ(n²) edges, all of which are critical. In the case of temporal cliques (the underlying graph is complete), spanners of size O(nlog n) are guaranteed. The original proof of this result by Casteigts et al. [ICALP 2019] combines a number of techniques, one of which is called dismountability. In a recent work, Angrick et al. [ESA 2024] simplified the proof and showed, among other things, that a one-sided version of dismountability can replace elegantly the second part of the proof. In this paper, we revisit methodically the dismountability principle. We start by characterizing the structure that a temporal clique must have if it is non 1-hop dismountable, then neither 1-hop nor 2-hop (i.e. non {1,2}-hop) dismountable, and finally non {1,2,3}-hop dismountable. It turns out that if a clique is k-hop dismountable for any other k, then it must also be {1,2,3}-hop dismountable, thus no additional structure can be obtained beyond this point. Interestingly, excluding 1-hop and 2-hop dismountability is already sufficient for reducing the spanner problem from cliques to extremally matched bicliques, where the O(nlog n) result is subsequently obtained. Put together with the strategy of Angrick et al., this entire result can now be recovered using only dismountability. An interesting by-product of our analysis is that any minimal counter-example to the existence of 4n spanners must satisfy the properties of non {1,2,3}-hop dismountable cliques. In the second part, we discuss further connections between dismountability and another technique called pivotability. In particular, we show that if a temporal clique is recursively k-hop dismountable, then it is also pivotable (and thus admits a 2n spanner, whatever k). We also study a family of labelings called full-range that forces both dismountability and pivotability. The latter gives some evidence that large lifetimes could be exploited more generally for the construction of spanners.
Cite as

Daniele Carnevale, Arnaud Casteigts, and Timothée Corsini. Dismountability in Temporal Cliques Revisited. In 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 330, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{carnevale_et_al:LIPIcs.SAND.2025.6,
  author =	{Carnevale, Daniele and Casteigts, Arnaud and Corsini, Timoth\'{e}e},
  title =	{{Dismountability in Temporal Cliques Revisited}},
  booktitle =	{4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-368-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{330},
  editor =	{Meeks, Kitty and Scheideler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2025.6},
  URN =		{urn:nbn:de:0030-drops-230591},
  doi =		{10.4230/LIPIcs.SAND.2025.6},
  annote =	{Keywords: Dynamic networks, Temporal graphs, Reachability, Dismountability, Pivotability, Temporal spanners, Full-range graphs}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.24
A Deterministic Approach to Shortest Path Restoration in Edge Faulty Graphs

Authors: Keerti Choudhary and Rishabh Dhiman

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
Afek, Bremler-Barr, Kaplan, Cohen, and Merritt (PODC'01) in their seminal work on shortest path restorations demonstrated that after a single edge failure in a graph G, a replacement shortest path between any two vertices s and t, which avoids the failed edge, can be represented as the concatenation of two original shortest paths in G. They also showed that we cannot associate a canonical shortest path between the vertex pairs in G that consistently allows for the replacement path (in the surviving graph) to be represented as a concatenation of these canonical paths. Recently, Bodwin and Parter (PODC'21) proposed a randomized tie-breaking scheme for selecting canonical paths for the "ordered" vertex pairs in graph G with the desired property of representing the replacement shortest path as a concatenation of canonical shortest-paths provided for ordered pairs. An interesting open question is whether it is possible to provide a deterministic construction of canonical paths in an efficient manner. We address this question in our paper by presenting an O(mn) time deterministic algorithm to compute a canonical path family ℱ = {P_{x,y}, Q_{x,y} | x,y ∈ V} comprising of two paths per (unordered) vertex pair. Each replacement is either a PQ-path (of type P_{x,y}∘Q_{y,z}), a QP-path, a QQ-path, or a PP-path. Our construction is fairly simple and is a straightforward application of independent spanning trees. We also present various applications of family ℱ in computing fault-tolerant structures.
Cite as

Keerti Choudhary and Rishabh Dhiman. A Deterministic Approach to Shortest Path Restoration in Edge Faulty Graphs. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 24:1-24:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{choudhary_et_al:LIPIcs.STACS.2025.24,
  author =	{Choudhary, Keerti and Dhiman, Rishabh},
  title =	{{A Deterministic Approach to Shortest Path Restoration in Edge Faulty Graphs}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{24:1--24:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.24},
  URN =		{urn:nbn:de:0030-drops-228499},
  doi =		{10.4230/LIPIcs.STACS.2025.24},
  annote =	{Keywords: Fault-tolerant Data-structures, Shortest Path Restoration, Replacement path}
}
Document
DOI: 10.4230/LIPIcs.SEA.2023.10
Solving Directed Feedback Vertex Set by Iterative Reduction to Vertex Cover

Authors: Sebastian Angrick, Ben Bals, Katrin Casel, Sarel Cohen, Tobias Friedrich, Niko Hastrich, Theresa Hradilak, Davis Issac, Otto Kißig, Jonas Schmidt, and Leo Wendt

Published in: LIPIcs, Volume 265, 21st International Symposium on Experimental Algorithms (SEA 2023)

Abstract
In the Directed Feedback Vertex Set (DFVS) problem, one is given a directed graph G = (V,E) and wants to find a minimum cardinality set S ⊆ V such that G-S is acyclic. DFVS is a fundamental problem in computer science and finds applications in areas such as deadlock detection. The problem was the subject of the 2022 PACE coding challenge. We develop a novel exact algorithm for the problem that is tailored to perform well on instances that are mostly bi-directed. For such instances, we adapt techniques from the well-researched vertex cover problem. Our core idea is an iterative reduction to vertex cover. To this end, we also develop a new reduction rule that reduces the number of not bi-directed edges. With the resulting algorithm, we were able to win third place in the exact track of the PACE challenge. We perform computational experiments and compare the running time to other exact algorithms, in particular to the winning algorithm in PACE. Our experiments show that we outpace the other algorithms on instances that have a low density of uni-directed edges.
Cite as

Sebastian Angrick, Ben Bals, Katrin Casel, Sarel Cohen, Tobias Friedrich, Niko Hastrich, Theresa Hradilak, Davis Issac, Otto Kißig, Jonas Schmidt, and Leo Wendt. Solving Directed Feedback Vertex Set by Iterative Reduction to Vertex Cover. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{angrick_et_al:LIPIcs.SEA.2023.10,
  author =	{Angrick, Sebastian and Bals, Ben and Casel, Katrin and Cohen, Sarel and Friedrich, Tobias and Hastrich, Niko and Hradilak, Theresa and Issac, Davis and Ki{\ss}ig, Otto and Schmidt, Jonas and Wendt, Leo},
  title =	{{Solving Directed Feedback Vertex Set by Iterative Reduction to Vertex Cover}},
  booktitle =	{21st International Symposium on Experimental Algorithms (SEA 2023)},
  pages =	{10:1--10:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-279-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{265},
  editor =	{Georgiadis, Loukas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.10},
  URN =		{urn:nbn:de:0030-drops-183602},
  doi =		{10.4230/LIPIcs.SEA.2023.10},
  annote =	{Keywords: directed feedback vertex set, vertex cover, reduction rules}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.24
Fault-Tolerant ST-Diameter Oracles

Authors: Davide Bilò, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, Simon Krogmann, and Martin Schirneck

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
We study the problem of estimating the ST-diameter of a graph that is subject to a bounded number of edge failures. An f-edge fault-tolerant ST-diameter oracle (f-FDO-ST) is a data structure that preprocesses a given graph G, two sets of vertices S,T, and positive integer f. When queried with a set F of at most f edges, the oracle returns an estimate D̂ of the ST-diameter diam(G-F,S,T), the maximum distance between vertices in S and T in G-F. The oracle has stretch σ ⩾ 1 if diam(G-F,S,T) ⩽ D̂ ⩽ σ diam(G-F,S,T). If S and T both contain all vertices, the data structure is called an f-edge fault-tolerant diameter oracle (f-FDO). An f-edge fault-tolerant distance sensitivity oracles (f-DSO) estimates the pairwise graph distances under up to f failures. We design new f-FDOs and f-FDO-STs by reducing their construction to that of all-pairs and single-source f-DSOs. We obtain several new tradeoffs between the size of the data structure, stretch guarantee, query and preprocessing times for diameter oracles by combining our black-box reductions with known results from the literature. We also provide an information-theoretic lower bound on the space requirement of approximate f-FDOs. We show that there exists a family of graphs for which any f-FDO with sensitivity f ⩾ 2 and stretch less than 5/3 requires Ω(n^{3/2}) bits of space, regardless of the query time.
Cite as

Davide Bilò, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, Simon Krogmann, and Martin Schirneck. Fault-Tolerant ST-Diameter Oracles. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bilo_et_al:LIPIcs.ICALP.2023.24,
  author =	{Bil\`{o}, Davide and Choudhary, Keerti and Cohen, Sarel and Friedrich, Tobias and Krogmann, Simon and Schirneck, Martin},
  title =	{{Fault-Tolerant ST-Diameter Oracles}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{24:1--24:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel 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.2023.24},
  URN =		{urn:nbn:de:0030-drops-180762},
  doi =		{10.4230/LIPIcs.ICALP.2023.24},
  annote =	{Keywords: diameter oracles, distance sensitivity oracles, space lower bounds, fault-tolerant data structures}
}
Document
PACE Solver Description
DOI: 10.4230/LIPIcs.IPEC.2022.28
PACE Solver Description: Mount Doom - An Exact Solver for Directed Feedback Vertex Set

Authors: Sebastian Angrick, Ben Bals, Katrin Casel, Sarel Cohen, Tobias Friedrich, Niko Hastrich, Theresa Hradilak, Davis Issac, Otto Kißig, Jonas Schmidt, and Leo Wendt

Published in: LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

Abstract
In this document we describe the techniques we used and implemented for our submission to the Parameterized Algorithms and Computational Experiments Challenge (PACE) 2022. The given problem is Directed Feedback Vertex Set (DFVS), where one is given a directed graph G = (V,E) and wants to find a minimum S ⊆ V such that G-S is acyclic. We approach this problem by first exhaustively applying a set of reduction rules. In order to find a minimum DFVS on the remaining instance, we create and solve a series of Vertex Cover instances.
Cite as

Sebastian Angrick, Ben Bals, Katrin Casel, Sarel Cohen, Tobias Friedrich, Niko Hastrich, Theresa Hradilak, Davis Issac, Otto Kißig, Jonas Schmidt, and Leo Wendt. PACE Solver Description: Mount Doom - An Exact Solver for Directed Feedback Vertex Set. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 28:1-28:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{angrick_et_al:LIPIcs.IPEC.2022.28,
  author =	{Angrick, Sebastian and Bals, Ben and Casel, Katrin and Cohen, Sarel and Friedrich, Tobias and Hastrich, Niko and Hradilak, Theresa and Issac, Davis and Ki{\ss}ig, Otto and Schmidt, Jonas and Wendt, Leo},
  title =	{{PACE Solver Description: Mount Doom - An Exact Solver for Directed Feedback Vertex Set}},
  booktitle =	{17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
  pages =	{28:1--28:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-260-0},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{249},
  editor =	{Dell, Holger and Nederlof, Jesper},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.28},
  URN =		{urn:nbn:de:0030-drops-173847},
  doi =		{10.4230/LIPIcs.IPEC.2022.28},
  annote =	{Keywords: directed feedback vertex set, vertex cover, reduction rules}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.22
Deterministic Sensitivity Oracles for Diameter, Eccentricities and All Pairs Distances

Authors: Davide Bilò, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, and Martin Schirneck

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
We construct data structures for extremal and pairwise distances in directed graphs in the presence of transient edge failures. Henzinger et al. [ITCS 2017] initiated the study of fault-tolerant (sensitivity) oracles for the diameter and vertex eccentricities. We extend this with a special focus on space efficiency. We present several new data structures, among them the first fault-tolerant eccentricity oracle for dual failures in subcubic space. We further prove lower bounds that show limits to approximation vs. space and diameter vs. space trade-offs for fault-tolerant oracles. They highlight key differences between data structures for undirected and directed graphs. Initially, our oracles are randomized leaning on a sampling technique frequently used in sensitivity analysis. Building on the work of Alon, Chechik, and Cohen [ICALP 2019] as well as Karthik and Parter [SODA 2021], we develop a hierarchical framework to derandomize fault-tolerant data structures. We first apply it to our own diameter and eccentricity oracles and then show its versatility by derandomizing algorithms from the literature: the distance sensitivity oracle of Ren [JCSS 2022] and the Single-Source Replacement Path algorithm of Chechik and Magen [ICALP 2020]. This way, we obtain the first deterministic distance sensitivity oracle with subcubic preprocessing time.
Cite as

Davide Bilò, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Deterministic Sensitivity Oracles for Diameter, Eccentricities and All Pairs Distances. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.ICALP.2022.22,
  author =	{Bil\`{o}, Davide and Choudhary, Keerti and Cohen, Sarel and Friedrich, Tobias and Schirneck, Martin},
  title =	{{Deterministic Sensitivity Oracles for Diameter, Eccentricities and All Pairs Distances}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{22:1--22:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.22},
  URN =		{urn:nbn:de:0030-drops-163633},
  doi =		{10.4230/LIPIcs.ICALP.2022.22},
  annote =	{Keywords: derandomization, diameter, eccentricity, fault-tolerant data structure, sensitivity oracle, space lower bound}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.23
Fixed-Parameter Sensitivity Oracles

Authors: Davide Bilò, Katrin Casel, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, J.A. Gregor Lagodzinski, Martin Schirneck, and Simon Wietheger

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
We combine ideas from distance sensitivity oracles (DSOs) and fixed-parameter tractability (FPT) to design sensitivity oracles for FPT graph problems. An oracle with sensitivity f for an FPT problem Π on a graph G with parameter k preprocesses G in time O(g(f,k) ⋅ poly(n)). When queried with a set F of at most f edges of G, the oracle reports the answer to the Π - with the same parameter k - on the graph G-F, i.e., G deprived of F. The oracle should answer queries in a time that is significantly faster than merely running the best-known FPT algorithm on G-F from scratch. We design sensitivity oracles for the k-Path and the k-Vertex Cover problem. Our first oracle for k-Path has size O(k^{f+1}) and query time O(f min{f, log(f) + k}). We use a technique inspired by the work of Weimann and Yuster [FOCS 2010, TALG 2013] on distance sensitivity problems to reduce the space to O(({f+k}/f)^f ({f+k}/k)^k fk⋅log(n)) at the expense of increasing the query time to O(({f+k}/f)^f ({f+k}/k)^k f min{f,k}⋅log(n)). Both oracles can be modified to handle vertex-failures, but we need to replace k with 2k in all the claimed bounds. Regarding k-Vertex Cover, we design three oracles offering different trade-offs between the size and the query time. The first oracle takes O(3^{f+k}) space and has O(2^f) query time, the second one has a size of O(2^{f+k²+k}) and a query time of O(f+k²); finally, the third one takes O(fk+k²) space and can be queried in time O(1.2738^k + f). All our oracles are computable in time (at most) proportional to their size and the time needed to detect a k-path or k-vertex cover, respectively. We also provide an interesting connection between k-Vertex Cover and the fault-tolerant shortest path problem, by giving a DSO of size O(poly(f,k) ⋅ n) with query time in O(poly(f,k)), where k is the size of a vertex cover. Following our line of research connecting fault-tolerant FPT and shortest paths problems, we introduce parameterization to the computation of distance preservers. We study the problem, given a directed unweighted graph with a fixed source s and parameters f and k, to construct a polynomial-sized oracle that efficiently reports, for any target vertex v and set F of at most f edges, whether the distance from s to v increases at most by an additive term of k in G-F. The oracle size is O(2^k k²⋅n), while the time needed to answer a query is O(2^k f^ω k^ω), where ω < 2.373 is the matrix multiplication exponent. The second problem we study is about the construction of bounded-stretch fault-tolerant preservers. We construct a subgraph with O(2^{fk+f+k} k ⋅ n) edges that preserves those s-v-distances that do not increase by more than k upon failure of F. This improves significantly over the Õ(f n^{2-1/(2^f)}) bound in the unparameterized case by Bodwin et al. [ICALP 2017].
Cite as

Davide Bilò, Katrin Casel, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, J.A. Gregor Lagodzinski, Martin Schirneck, and Simon Wietheger. Fixed-Parameter Sensitivity Oracles. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.ITCS.2022.23,
  author =	{Bil\`{o}, Davide and Casel, Katrin and Choudhary, Keerti and Cohen, Sarel and Friedrich, Tobias and Lagodzinski, J.A. Gregor and Schirneck, Martin and Wietheger, Simon},
  title =	{{Fixed-Parameter Sensitivity Oracles}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{23:1--23:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.23},
  URN =		{urn:nbn:de:0030-drops-156196},
  doi =		{10.4230/LIPIcs.ITCS.2022.23},
  annote =	{Keywords: data structures, distance preservers, distance sensitivity oracles, fault tolerance, fixed-parameter tractability, k-path, vertex cover}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.18
Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles

Authors: Davide Bilò, Sarel Cohen, Tobias Friedrich, and Martin Schirneck

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract
Given a graph with a distinguished source vertex s, the Single Source Replacement Paths (SSRP) problem is to compute and output, for any target vertex t and edge e, the length d(s,t,e) of a shortest path from s to t that avoids a failing edge e. A Single-Source Distance Sensitivity Oracle (Single-Source DSO) is a compact data structure that answers queries of the form (t,e) by returning the distance d(s,t,e). We show how to deterministically compress the output of the SSRP problem on n-vertex, m-edge graphs with integer edge weights in the range [1,M] into a Single-Source DSO that has size O(M^{1/2} n^{3/2}) and query time Õ(1). We prove that the space requirement is optimal (up to the word size). Our techniques can also handle vertex failures within the same bounds. Chechik and Cohen [SODA 2019] presented a combinatorial, randomized Õ(m√n+n²) time SSRP algorithm for undirected and unweighted graphs. We derandomize their algorithm with the same asymptotic running time and apply our compression to obtain a deterministic Single-Source DSO with Õ(m√n+n²) preprocessing time, O(n^{3/2}) space, and Õ(1) query time. Our combinatorial Single-Source DSO has near-optimal space, preprocessing and query time for unweighted graphs, improving the preprocessing time by a √n-factor compared to previous results with o(n²) space. Grandoni and Vassilevska Williams [FOCS 2012, TALG 2020] gave an algebraic, randomized Õ(Mn^ω) time SSRP algorithm for (undirected and directed) graphs with integer edge weights in the range [1,M], where ω < 2.373 is the matrix multiplication exponent. We derandomize it for undirected graphs and apply our compression to obtain an algebraic Single-Source DSO with Õ(Mn^ω) preprocessing time, O(M^{1/2} n^{3/2}) space, and Õ(1) query time. This improves the preprocessing time of algebraic Single-Source DSOs by polynomial factors compared to previous o(n²)-space oracles. We also present further improvements of our Single-Source DSOs. We show that the query time can be reduced to a constant at the cost of increasing the size of the oracle to O(M^{1/3} n^{5/3}) and that all our oracles can be made path-reporting. On sparse graphs with m = O(n^{5/4-ε}/M^{7/4}) edges, for any constant ε > 0, we reduce the preprocessing to randomized Õ(M^{7/8} m^{1/2} n^{11/8}) = O(n^{2-ε/2}) time. To the best of our knowledge, this is the first truly subquadratic time algorithm for building Single-Source DSOs on sparse graphs.
Cite as

Davide Bilò, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bilo_et_al:LIPIcs.ESA.2021.18,
  author =	{Bil\`{o}, Davide and Cohen, Sarel and Friedrich, Tobias and Schirneck, Martin},
  title =	{{Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus 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.2021.18},
  URN =		{urn:nbn:de:0030-drops-145999},
  doi =		{10.4230/LIPIcs.ESA.2021.18},
  annote =	{Keywords: derandomization, distance sensitivity oracle, single-source replacement paths, space lower bound}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.18
Space-Efficient Fault-Tolerant Diameter Oracles

Authors: Davide Bilò, Sarel Cohen, Tobias Friedrich, and Martin Schirneck

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract
We design f-edge fault-tolerant diameter oracles (f-FDO, or simply FDO if f = 1). For a given directed or undirected and possibly edge-weighted graph G with n vertices and m edges and a positive integer f, we preprocess the graph and construct a data structure that, when queried with a set F of edges, where |F| ⩽ f, returns the diameter of G-F. An f-FDO has stretch σ ⩾ 1 if the returned value D^ satisfies diam(G-F) ⩽ D^ ⩽ σ diam(G-F). For the case of a single edge failure (f = 1) in an unweighted directed graph, there exists an approximate FDO by Henzinger et al. [ITCS 2017] with stretch (1+ε), constant query time, space O(m), and a combinatorial preprocessing time of Õ(mn + n^{1.5} √{Dm/ε}), where D is the diameter. We present an FDO for directed graphs with the same stretch, query time, and space. It has a preprocessing time of Õ(mn + n²/ε), which is better for constant ε > 0. The preprocessing time nearly matches a conditional lower bound for combinatorial algorithms, also by Henzinger et al. With fast matrix multiplication, we achieve a preprocessing time of Õ(n^{2.5794} + n²/ε). We further prove an information-theoretic lower bound showing that any FDO with stretch better than 3/2 requires Ω(m) bits of space. Thus, for constant 0 < ε < 3/2, our combinatorial (1+ε)-approximate FDO is near-optimal in all parameters. In the case of multiple edge failures (f > 1) in undirected graphs with non-negative edge weights, we give an f-FDO with stretch (f+2), query time O(f²log²{n}), Õ(fn) space, and preprocessing time Õ(fm). We complement this with a lower bound excluding any finite stretch in o(fn) space. Many real-world networks have polylogarithmic diameter. We show that for those graphs and up to f = o(log n/ log log n) failures one can swap approximation for query time and space. We present an exact combinatorial f-FDO with preprocessing time mn^{1+o(1)}, query time n^o(1), and space n^{2+o(1)}. When using fast matrix multiplication instead, the preprocessing time can be improved to n^{ω+o(1)}, where ω < 2.373 is the matrix multiplication exponent.
Cite as

Davide Bilò, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Space-Efficient Fault-Tolerant Diameter Oracles. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bilo_et_al:LIPIcs.MFCS.2021.18,
  author =	{Bil\`{o}, Davide and Cohen, Sarel and Friedrich, Tobias and Schirneck, Martin},
  title =	{{Space-Efficient Fault-Tolerant Diameter Oracles}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{18:1--18:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.18},
  URN =		{urn:nbn:de:0030-drops-144581},
  doi =		{10.4230/LIPIcs.MFCS.2021.18},
  annote =	{Keywords: derandomization, diameter, distance sensitivity oracle, fault-tolerant data structure, space lower bound}
}
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