Search Results

Documents authored by Roditty, Liam

Document
DOI: 10.4230/LIPIcs.FSTTCS.2025.50
Additive, Near-Additive, and Multiplicative Approximations for APSP in Weighted Undirected Graphs: Trade-Offs and Algorithms

Authors: Liam Roditty and Ariel Sapir

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

Abstract
We present a +2∑_{i=1}^{k+1} W_i-APASP algorithm for dense weighted graphs with a runtime of Õ(n^{2+1/(3k+2)), where W_i is the weight of an i^th heaviest edge on a shortest path between two vertices. Dor, Halperin and Zwick [FOCS'96 and SICOMP'00] introduced two algorithms for the commensurate unweighted +2⋅ (k+1)-APASP problem: one for sparse graphs with a runtime of Õ(n^{2-1/(k+2)} m^{1/(k+2)}) and one for dense graphs with a runtime of Õ(n^{2+1/(3k+2)}). Subsequently, Cohen and Zwick [SODA'97 and JALG'01] adapted the algorithm for sparse graphs to the weighted setting, namely a +2∑_{i=1}^{k+1} W_i-APASP algorithm with the same Õ(n^{2-1/(k+2)} m^{1/(k+2)}) runtime. We fill the nearly three decades old gap by providing an algorithm for dense weighted graphs, matching the runtime for the unweighted setting. In addition, we explore nearly additive APASP, where the multiplicative stretch is 1+ε. We present a (1+ε, min{2W₁,4W₂})-APASP algorithm with a runtime of Õ((1/ε)^{O(1)} ⋅ n^{2.15135313} ⋅ log W). This improves upon Saha and Ye [SODA'24], which had the same runtime, yet (1+ε, 2W₁)-APASP. For pure multiplicative APASP, we present a (7/3+ε)-APASP algorithm with a runtime of Õ((1/ε)^{O(1)} ⋅ n^{2.15135313} ⋅ log W). This improves, for dense graphs, the Õ(nm^{2/3}+n²) runtime of the 7/3-APASP algorithm by Baswana and Kavitha [FOCS'06 and SICOMP'10], at the cost of introducing an additional ε to the multiplicative stretch. We further view this result in a broader framework of ((3𝓁+4)/(𝓁+2) + ε)-APASP algorithms, similarly to the family of (3𝓁+4)/(𝓁+2)-APASP algorithms by Akav and Roditty [ESA'21]. This also generalizes the (2+ε)-APASP algorithm by Dory, Forster, Kirkpatrick, Nazari, Vassilevska Williams, and de Vos [SODA'24]. Finally, we show that it is possible to "bypass" an Ω̃ (n^ω) conditional lower bound by Dor, Halperin, and Zwick for α-APASP with α < 2, by allowing an additive component to the approximation (e.g. a ((6k+3)/(3k+2),∑_{i=1}^{k+1} W_i)-APASP with Õ(n^{2+1/(3k+2)}) runtime.).
Cite as

Liam Roditty and Ariel Sapir. Additive, Near-Additive, and Multiplicative Approximations for APSP in Weighted Undirected Graphs: Trade-Offs and Algorithms. 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. 50:1-50:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{roditty_et_al:LIPIcs.FSTTCS.2025.50,
  author =	{Roditty, Liam and Sapir, Ariel},
  title =	{{Additive, Near-Additive, and Multiplicative Approximations for APSP in Weighted Undirected Graphs: Trade-Offs and Algorithms}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{50:1--50:22},
  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.2025.50},
  URN =		{urn:nbn:de:0030-drops-251309},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.50},
  annote =	{Keywords: Graph, Shortest Paths, Weighted Graphs, Approximation, Undirected, Single Source Shortest-Paths, Multi-Source Shortest-Paths, All-Pairs Shortest-Paths, SSSP, MSSP, MSASP, APSP, APASP}
}
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)

Copy BibTex To Clipboard

@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.2025.38
Compact Routing Schemes in Undirected and Directed Graphs

Authors: Avi Kadria and Liam Roditty

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

Abstract
In this paper, we study the problem of compact routing schemes in weighted undirected and directed graphs. For weighted undirected graphs, more than a decade ago, Chechik [PODC'13] presented a ≈ 3.68k-stretch compact routing scheme that uses Õ(n^{1/k}log{D}) local storage, where D is the normalized diameter, for every k > 1. We present a ≈ 2.64k-stretch compact routing scheme that uses Õ(n^{1/k}) local storage on average in each vertex. This is the first compact routing scheme that uses total local storage of Õ(n^{1+1/k}) while achieving a c ⋅ k stretch, for a constant c < 3. In real-world network protocols, messages are usually transmitted as part of a communication session between two parties. Therefore, more than two decades ago, Thorup and Zwick [SPAA'01] considered compact routing schemes that establish a communication session using a handshake. In their handshake-based compact routing scheme, the handshake is routed along a (4k-5)-stretch path, and the rest of the communication session is routed along an optimal (2k-1)-stretch path. It is straightforward to improve the (4k-5)-stretch of the handshake to ≈ 3.68k-stretch using the compact routing scheme of Chechik [PODC'13]. We improve the handshake stretch to the optimal (2k-1), by borrowing the concept of roundtrip routing from directed graphs to undirected graphs. For weighted directed graphs, more than two decades ago, Roditty, Thorup, and Zwick [SODA'02 and TALG'08] presented a (4k+ε)-stretch compact roundtrip routing scheme that uses Õ(n^{1/k}) local storage for every k ≥ 3. For k = 3, this gives a (12+ε)-roundtrip stretch using Õ(n^{1/3}) local storage. We improve the stretch by developing a 7-roundtrip stretch routing scheme with Õ(n^{1/3}) local storage. In addition, we consider graphs with bounded hop diameter and present an optimal (2k-1)-roundtrip stretch routing scheme that uses Õ(D_{HOP}⋅ n^{1/k}), where D_{HOP} is the hop diameter of the graph.
Cite as

Avi Kadria and Liam Roditty. Compact Routing Schemes in Undirected and Directed Graphs. In 39th International Symposium on Distributed Computing (DISC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 356, pp. 38:1-38:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{kadria_et_al:LIPIcs.DISC.2025.38,
  author =	{Kadria, Avi and Roditty, Liam},
  title =	{{Compact Routing Schemes in Undirected and Directed Graphs}},
  booktitle =	{39th International Symposium on Distributed Computing (DISC 2025)},
  pages =	{38:1--38:19},
  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.2025.38},
  URN =		{urn:nbn:de:0030-drops-248555},
  doi =		{10.4230/LIPIcs.DISC.2025.38},
  annote =	{Keywords: Routing schemes, Compact routing schemes, Distance oracles, Computer networks, Graph algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.101
On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch

Authors: Tsvi Kopelowitz, Ariel Korin, and Liam Roditty

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
For an undirected unweighted graph G = (V,E) with n vertices and m edges, let d(u,v) denote the distance from u ∈ V to v ∈ V in G. An (α,β)-stretch approximate distance oracle (ADO) for G is a data structure that given u,v ∈ V returns in constant (or near constant) time a value dˆ(u,v) such that d(u,v) ≤ dˆ(u,v) ≤ α⋅ d(u,v) + β, for some reals α > 1, β. Thorup and Zwick [Mikkel Thorup and Uri Zwick, 2005] showed that one cannot beat stretch 3 with subquadratic space (in terms of n) for general graphs. Pǎtraşcu and Roditty [Mihai Pǎtraşcu and Liam Roditty, 2010] showed that one can obtain stretch 2 using O(m^{1/3}n^{4/3}) space, and so if m is subquadratic in n then the space usage is also subquadratic. Moreover, Pǎtraşcu and Roditty [Mihai Pǎtraşcu and Liam Roditty, 2010] showed that one cannot beat stretch 2 with subquadratic space even for graphs where m = Õ(n), based on the set-intersection hypothesis. In this paper we explore the conditions for which an ADO can beat stretch 2 while using subquadratic space. In particular, we show that if the maximum degree in G is Δ_G ≤ O(n^{1/k-ε}) for some 0 < ε ≤ 1/k, then there exists an ADO for G that uses Õ(n^{2-(kε)/3) space and has a (2,1-k)-stretch. For k = 2 this result implies a subquadratic sub-2 stretch ADO for graphs with Δ_G ≤ O(n^{1/2-ε}). Moreover, we prove a conditional lower bound, based on the set intersection hypothesis, which states that for any positive integer k ≤ log n, obtaining a sub-(k+2)/k stretch for graphs with Δ_G = Θ(n^{1/k}) requires Ω̃(n²) space. Thus, for graphs with maximum degree Θ(n^{1/2}), obtaining a sub-2 stretch requires Ω̃(n²) space.
Cite as

Tsvi Kopelowitz, Ariel Korin, and Liam Roditty. On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 101:1-101:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{kopelowitz_et_al:LIPIcs.ICALP.2024.101,
  author =	{Kopelowitz, Tsvi and Korin, Ariel and Roditty, Liam},
  title =	{{On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{101:1--101:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.101},
  URN =		{urn:nbn:de:0030-drops-202443},
  doi =		{10.4230/LIPIcs.ICALP.2024.101},
  annote =	{Keywords: Graph algorithms, Approximate distance oracle, data structures, shortest path}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.49
Dynamic Connectivity in Disk Graphs

Authors: Haim Kaplan, Alexander Kauer, Katharina Klost, Kristin Knorr, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
Let S ⊆ ℝ² be a set of n planar sites, such that each s ∈ S has an associated radius r_s > 0. Let 𝒟(S) be the disk intersection graph for S. It has vertex set S and an edge between two distinct sites s, t ∈ S if and only if the disks with centers s, t and radii r_s, r_t intersect. Our goal is to design data structures that maintain the connectivity structure of 𝒟(S) as sites are inserted and/or deleted. First, we consider unit disk graphs, i.e., r_s = 1, for all s ∈ S. We describe a data structure that has O(log² n) amortized update and O(log n/log log n) amortized query time. Second, we look at disk graphs with bounded radius ratio Ψ, i.e., for all s ∈ S, we have 1 ≤ r_s ≤ Ψ, for a Ψ ≥ 1 known in advance. In the fully dynamic case, we achieve amortized update time O(Ψ λ₆(log n) log⁷ n) and query time O(log n/log log n), where λ_s(n) is the maximum length of a Davenport-Schinzel sequence of order s on n symbols. In the incremental case, where only insertions are allowed, we get logarithmic dependency on Ψ, with O(α(n)) query time and O(logΨ λ₆(log n) log⁷ n) update time. For the decremental setting, where only deletions are allowed, we first develop an efficient disk revealing structure: given two sets R and B of disks, we can delete disks from R, and upon each deletion, we receive a list of all disks in B that no longer intersect the union of R. Using this, we get decremental data structures with amortized query time O(log n/log log n) that support m deletions in O((nlog⁵ n + m log⁷ n) λ₆(log n) + nlog Ψ log⁴n) overall time for bounded radius ratio Ψ and O((nlog⁶ n + m log⁸n) λ₆(log n)) for arbitrary radii.
Cite as

Haim Kaplan, Alexander Kauer, Katharina Klost, Kristin Knorr, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth. Dynamic Connectivity in Disk Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 49:1-49:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{kaplan_et_al:LIPIcs.SoCG.2022.49,
  author =	{Kaplan, Haim and Kauer, Alexander and Klost, Katharina and Knorr, Kristin and Mulzer, Wolfgang and Roditty, Liam and Seiferth, Paul},
  title =	{{Dynamic Connectivity in Disk Graphs}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{49:1--49:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.49},
  URN =		{urn:nbn:de:0030-drops-160572},
  doi =		{10.4230/LIPIcs.SoCG.2022.49},
  annote =	{Keywords: Disk Graphs, Connectivity, Lower Envelopes}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.4
A Unified Approach for All Pairs Approximate Shortest Paths in Weighted Undirected Graphs

Authors: Maor Akav and Liam Roditty

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

Abstract
Let G = (V,E) be a weighted undirected graph with n vertices and m edges, and let d_G(u,v) be the length of the shortest path between u and v in G. In this paper we present a unified approach for obtaining algorithms for all pairs approximate shortest paths in weighted undirected graphs. For every integer k ≥ 2 we show that there is an Õ(n²+kn^{2-3/k}m^{2/k}) expected running time algorithm that computes a matrix M such that for every u,v ∈ V: d_G(u,v) ≤ M[u,v] ≤ (2+(k-2)/k)d_G(u,v). Previous algorithms obtained only specific approximation factors. Baswana and Kavitha [FOCS 2006, SICOMP 2010] presented a 2-approximation algorithm with expected running time of Õ(n²+m√ n) and a 7/3-approximation algorithm with expected running time of Õ(n²+m^{2/3}n). Their results improved upon the results of Cohen and Zwick [SODA 1997, JoA 2001] for graphs with m = o(n²). Kavitha [FSTTCS 2007, Algorithmica 2012] presented a 5/2-approximation algorithm with expected running time of Õ(n^{9/4}). For k = 2 and k = 3 our result gives the algorithms of Baswana and Kavitha. For k = 4, we get a 5/2-approximation algorithm with Õ(n^{5/4}m^{1/2}) expected running time. This improves upon the running time of Õ(n^{9/4}) due to Kavitha, when m = o(n²). Our unified approach reveals that all previous algorithms are a part of a family of algorithms that exhibit a smooth tradeoff between approximation of 2 and 3, and are not sporadic unrelated results. Moreover, our new algorithm uses, among other ideas, the celebrated approximate distance oracles of Thorup and Zwick [STOC 2001, JACM 2005] in a non standard way, which we believe is of independent interest, due to their extensive usage in a variety of applications.
Cite as

Maor Akav and Liam Roditty. A Unified Approach for All Pairs Approximate Shortest Paths in Weighted Undirected Graphs. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{akav_et_al:LIPIcs.ESA.2021.4,
  author =	{Akav, Maor and Roditty, Liam},
  title =	{{A Unified Approach for All Pairs Approximate Shortest Paths in Weighted Undirected Graphs}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{4:1--4:18},
  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.2021.4},
  URN =		{urn:nbn:de:0030-drops-145858},
  doi =		{10.4230/LIPIcs.ESA.2021.4},
  annote =	{Keywords: Graph algorithms, Approximate All Pairs of Shortest Paths, Distance Oracles}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.64
Triangles and Girth in Disk Graphs and Transmission Graphs

Authors: Haim Kaplan, Katharina Klost, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
Let S subset R^2 be a set of n sites, where each s in S has an associated radius r_s > 0. The disk graph D(S) is the undirected graph with vertex set S and an undirected edge between two sites s, t in S if and only if |st| <= r_s + r_t, i.e., if the disks with centers s and t and respective radii r_s and r_t intersect. Disk graphs are used to model sensor networks. Similarly, the transmission graph T(S) is the directed graph with vertex set S and a directed edge from a site s to a site t if and only if |st| <= r_s, i.e., if t lies in the disk with center s and radius r_s. We provide algorithms for detecting (directed) triangles and, more generally, computing the length of a shortest cycle (the girth) in D(S) and in T(S). These problems are notoriously hard in general, but better solutions exist for special graph classes such as planar graphs. We obtain similarly efficient results for disk graphs and for transmission graphs. More precisely, we show that a shortest (Euclidean) triangle in D(S) and in T(S) can be found in O(n log n) expected time, and that the (weighted) girth of D(S) can be found in O(n log n) expected time. For this, we develop new tools for batched range searching that may be of independent interest.
Cite as

Haim Kaplan, Katharina Klost, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Triangles and Girth in Disk Graphs and Transmission Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 64:1-64:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{kaplan_et_al:LIPIcs.ESA.2019.64,
  author =	{Kaplan, Haim and Klost, Katharina and Mulzer, Wolfgang and Roditty, Liam and Seiferth, Paul and Sharir, Micha},
  title =	{{Triangles and Girth in Disk Graphs and Transmission Graphs}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{64:1--64:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola 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.2019.64},
  URN =		{urn:nbn:de:0030-drops-111859},
  doi =		{10.4230/LIPIcs.ESA.2019.64},
  annote =	{Keywords: disk graph, transmission graph, triangle, girth}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.13
Algorithms and Hardness for Diameter in Dynamic Graphs

Authors: Bertie Ancona, Monika Henzinger, Liam Roditty, Virginia Vassilevska Williams, and Nicole Wein

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
The diameter, radius and eccentricities are natural graph parameters. While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each update or from solving dynamic All-Pairs Shortest Paths (APSP), which is very computationally intensive. This is the situation for dynamic approximation algorithms as well, and even if only edge insertions or edge deletions need to be supported. This paper provides a comprehensive study of the dynamic approximation of Diameter, Radius and Eccentricities, providing both conditional lower bounds, and new algorithms whose bounds are optimal under popular hypotheses in fine-grained complexity. Some of the highlights include: - Under popular hardness hypotheses, there can be no significantly better fully dynamic approximation algorithms than recomputing the answer after each update, or maintaining full APSP. - Nearly optimal partially dynamic (incremental/decremental) algorithms can be achieved via efficient reductions to (incremental/decremental) maintenance of Single-Source Shortest Paths. For instance, a nearly (3/2+epsilon)-approximation to Diameter in directed or undirected n-vertex, m-edge graphs can be maintained decrementally in total time m^{1+o(1)}sqrt{n}/epsilon^2. This nearly matches the static 3/2-approximation algorithm for the problem that is known to be conditionally optimal.
Cite as

Bertie Ancona, Monika Henzinger, Liam Roditty, Virginia Vassilevska Williams, and Nicole Wein. Algorithms and Hardness for Diameter in Dynamic Graphs. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{ancona_et_al:LIPIcs.ICALP.2019.13,
  author =	{Ancona, Bertie and Henzinger, Monika and Roditty, Liam and Williams, Virginia Vassilevska and Wein, Nicole},
  title =	{{Algorithms and Hardness for Diameter in Dynamic Graphs}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{13:1--13:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.13},
  URN =		{urn:nbn:de:0030-drops-105891},
  doi =		{10.4230/LIPIcs.ICALP.2019.13},
  annote =	{Keywords: fine-grained complexity, graph algorithms, dynamic algorithms}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2018.50
Stabbing Pairwise Intersecting Disks by Five Points

Authors: Sariel Har-Peled, Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, Micha Sharir, and Max Willert

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Abstract
Suppose we are given a set D of n pairwise intersecting disks in the plane. A planar point set P stabs D if and only if each disk in D contains at least one point from P. We present a deterministic algorithm that takes O(n) time to find five points that stab D. Furthermore, we give a simple example of 13 pairwise intersecting disks that cannot be stabbed by three points. This provides a simple - albeit slightly weaker - algorithmic version of a classical result by Danzer that such a set D can always be stabbed by four points.
Cite as

Sariel Har-Peled, Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, Micha Sharir, and Max Willert. Stabbing Pairwise Intersecting Disks by Five Points. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 50:1-50:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{harpeled_et_al:LIPIcs.ISAAC.2018.50,
  author =	{Har-Peled, Sariel and Kaplan, Haim and Mulzer, Wolfgang and Roditty, Liam and Seiferth, Paul and Sharir, Micha and Willert, Max},
  title =	{{Stabbing Pairwise Intersecting Disks by Five Points}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{50:1--50:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.50},
  URN =		{urn:nbn:de:0030-drops-99989},
  doi =		{10.4230/LIPIcs.ISAAC.2018.50},
  annote =	{Keywords: Disk graph, piercing set, LP-type problem}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2017.72
An Efficient Strongly Connected Components Algorithm in the Fault Tolerant Model

Authors: Surender Baswana, Keerti Choudhary, and Liam Roditty

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Abstract
In this paper we study the problem of maintaining the strongly connected components of a graph in the presence of failures. In particular, we show that given a directed graph G=(V,E) with n=|V| and m=|E|, and an integer value k\geq 1, there is an algorithm that computes in O(2^{k}n log^2 n) time for any set F of size at most k the strongly connected components of the graph G\F. The running time of our algorithm is almost optimal since the time for outputting the SCCs of G\F is at least \Omega(n). The algorithm uses a data structure that is computed in a preprocessing phase in polynomial time and is of size O(2^{k} n^2). Our result is obtained using a new observation on the relation between strongly connected components (SCCs) and reachability. More specifically, one of the main building blocks in our result is a restricted variant of the problem in which we only compute strongly connected components that intersect a certain path. Restricting our attention to a path allows us to implicitly compute reachability between the path vertices and the rest of the graph in time that depends logarithmically rather than linearly in the size of the path. This new observation alone, however, is not enough, since we need to find an efficient way to represent the strongly connected components using paths. For this purpose we use a mixture of old and classical techniques such as the heavy path decomposition of Sleator and Tarjan and the classical Depth-First-Search algorithm. Although, these are by now standard techniques, we are not aware of any usage of them in the context of dynamic maintenance of SCCs. Therefore, we expect that our new insights and mixture of new and old techniques will be of independent interest.
Cite as

Surender Baswana, Keerti Choudhary, and Liam Roditty. An Efficient Strongly Connected Components Algorithm in the Fault Tolerant Model. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 72:1-72:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

Copy BibTex To Clipboard

@InProceedings{baswana_et_al:LIPIcs.ICALP.2017.72,
  author =	{Baswana, Surender and Choudhary, Keerti and Roditty, Liam},
  title =	{{An Efficient  Strongly Connected Components Algorithm in the Fault Tolerant Model}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{72:1--72:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.72},
  URN =		{urn:nbn:de:0030-drops-74168},
  doi =		{10.4230/LIPIcs.ICALP.2017.72},
  annote =	{Keywords: Fault tolerant, Directed graph, Strongly connected components}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.156
Spanners and Reachability Oracles for Directed Transmission Graphs

Authors: Haim Kaplan, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract
Let P be a set of n points in d dimensions, each with an associated radius r_p > 0. The transmission graph G for P has vertex set P and an edge from p to q if and only if q lies in the ball with radius r_p around p. Let t > 1. A t-spanner H for G is a sparse subgraph of G such that for any two vertices p, q connected by a path of length l in G, there is a p-q-path of length at most tl in H. We show how to compute a t-spanner for G if d=2. The running time is O(n (log n + log Psi)), where Psi is the ratio of the largest and smallest radius of two points in P. We extend this construction to be independent of Psi at the expense of a polylogarithmic overhead in the running time. As a first application, we prove a property of the t-spanner that allows us to find a BFS tree in G for any given start vertex s of P in the same time. After that, we deal with reachability oracles for G. These are data structures that answer reachability queries: given two vertices, is there a directed path between them? The quality of a reachability oracle is measured by the space S(n), the query time Q(n), and the preproccesing time. For d=1, we show how to compute an oracle with Q(n) = O(1) and S(n) = O(n) in time O(n log n). For d=2, the radius ratio Psi again turns out to be an important measure for the complexity of the problem. We present three different data structures whose quality depends on Psi: (i) if Psi < sqrt(3), we achieve Q(n) = O(1) with S(n) = O(n) and preproccesing time O(n log n); (ii) if Psi >= sqrt(3), we get Q(n) = O(Psi^3 sqrt(n)) and S(n) = O(Psi^5 n^(3/2)); and (iii) if Psi is polynomially bounded in n, we use probabilistic methods to obtain an oracle with Q(n) = O(n^(2/3)log n) and S(n) = O(n^(5/3) log n) that answers queries correctly with high probability. We employ our t-spanner to achieve a fast preproccesing time of O(Psi^5 n^(3/2)) and O(n^(5/3) log^2 n) in case (ii) and (iii), respectively.
Cite as

Haim Kaplan, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth. Spanners and Reachability Oracles for Directed Transmission Graphs. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 156-170, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

Copy BibTex To Clipboard

@InProceedings{kaplan_et_al:LIPIcs.SOCG.2015.156,
  author =	{Kaplan, Haim and Mulzer, Wolfgang and Roditty, Liam and Seiferth, Paul},
  title =	{{Spanners and Reachability Oracles for Directed Transmission Graphs}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{156--170},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.156},
  URN =		{urn:nbn:de:0030-drops-51062},
  doi =		{10.4230/LIPIcs.SOCG.2015.156},
  annote =	{Keywords: Transmission Graphs, Reachability Oracles, Spanner, Intersection Graph}
}
Document
DOI: 10.4230/LIPIcs.STACS.2010.2489
Relaxed Spanners for Directed Disk Graphs

Authors: David Peleg and Liam Roditty

Published in: LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Abstract
Let $(V,\delta)$ be a finite metric space, where $V$ is a set of $n$ points and $\delta$ is a distance function defined for these points. Assume that $(V,\delta)$ has a constant doubling dimension $d$ and assume that each point $p\in V$ has a disk of radius $r(p)$ around it. The disk graph that corresponds to $V$ and $r(\cdot)$ is a \emph{directed} graph $I(V,E,r)$, whose vertices are the points of $V$ and whose edge set includes a directed edge from $p$ to $q$ if $\delta(p,q)\leq r(p)$. In~\cite{PeRo08} we presented an algorithm for constructing a $(1+\eps)$-spanner of size $O(n/\eps^d \log M)$, where $M$ is the maximal radius $r(p)$. The current paper presents two results. The first shows that the spanner of~\cite{PeRo08} is essentially optimal, i.e., for metrics of constant doubling dimension it is not possible to guarantee a spanner whose size is independent of $M$. The second result shows that by slightly relaxing the requirements and allowing a small perturbation of the radius assignment, considerably better spanners can be constructed. In particular, we show that if it is allowed to use edges of the disk graph $I(V,E,r_{1+\eps})$, where $r_{1+\eps}(p) = (1+\eps)\cdot r(p)$ for every $p\in V$, then it is possible to get a $(1+\eps)$-spanner of size $O(n/\eps^d)$ for $I(V,E,r)$. Our algorithm is simple and can be implemented efficiently.
Cite as

David Peleg and Liam Roditty. Relaxed Spanners for Directed Disk Graphs. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 609-620, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

Copy BibTex To Clipboard

@InProceedings{peleg_et_al:LIPIcs.STACS.2010.2489,
  author =	{Peleg, David and Roditty, Liam},
  title =	{{Relaxed Spanners for Directed Disk Graphs}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{609--620},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-16-3},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{5},
  editor =	{Marion, Jean-Yves and Schwentick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2489},
  URN =		{urn:nbn:de:0030-drops-24898},
  doi =		{10.4230/LIPIcs.STACS.2010.2489},
  annote =	{Keywords: Spanners, directed graphs}
}
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact