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Documents authored by Kadria, Avi

Document
DOI: 10.4230/LIPIcs.ESA.2026.66
Improved Approximation Algorithms for n-Pairs Shortest Paths

Authors: Avi Kadria, Liam Roditty, and Virginia Vassilevska Williams

Published in: LIPIcs, Volume 388, 34th Annual European Symposium on Algorithms (ESA 2026)

Abstract
Let G = (V, E) be a graph with n = |V| nodes and m = |E| edges. The t-Pairs Shortest Paths problem, introduced by Cohen [FOCS'93; SICOMP'99], asks to approximate the distances between t prespecified pairs of vertices. Recently, this problem has received renewed attention, particularly in the case where t = Θ(n): the n-Pairs Shortest Paths problem. In this setting, new algorithms and conditional lower bounds have been developed by Dalirrooyfard, Jin, Vassilevska Williams, and Wein [FOCS'22], and Chechik, Hoch, and Lifshitz [SODA'25]. In this paper, we present the first algorithm for the n-Pairs Shortest Paths problem in weighted undirected graphs that achieves a (2 - α)k-approximation, for constant α > 0, that runs in Õ(mn^{1/k} + n^{1 + 2/k}) time. Specifically, we present a 1.622k-approximation, improving upon the (2k - 3)-approximation of Chechik, Hoch, and Lifshitz [SODA'25] for graphs that are not super sparse, which answers in the affirmative the open question posed by them. We also develop improved approximation algorithms with better tradeoffs for unweighted graphs and dense weighted graphs that improve upon the results of Dalirrooyfard et al. and Chechik, Hoch, and Lifshitz. Our main technical contribution is the new heavy-edge technique. Using this technique, we transform an algorithm with an approximation guarantee that depends on W_{uv}, the weight of the heaviest edge on the shortest path between u and v, into an algorithm with purely multiplicative approximation that does not depend on W_{uv}.
Cite as

Avi Kadria, Liam Roditty, and Virginia Vassilevska Williams. Improved Approximation Algorithms for n-Pairs Shortest Paths. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 66:1-66:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kadria_et_al:LIPIcs.ESA.2026.66,
  author =	{Kadria, Avi and Roditty, Liam and Vassilevska Williams, Virginia},
  title =	{{Improved Approximation Algorithms for n-Pairs Shortest Paths}},
  booktitle =	{34th Annual European Symposium on Algorithms (ESA 2026)},
  pages =	{66:1--66:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-445-1},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{388},
  editor =	{Bille, Philip and Pettie, Seth and Storandt, Sabine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.66},
  URN =		{urn:nbn:de:0030-drops-272028},
  doi =		{10.4230/LIPIcs.ESA.2026.66},
  annote =	{Keywords: Fine-grained complexity, Graph algorithm, Graph distances, n pairs shortest paths, all pairs shortest paths, distance oracles}
}
Document
DOI: 10.4230/LIPIcs.ESA.2026.82
Tighter Bounds for Weighted and Unweighted Shortest Cycle Approximation

Authors: Avi Kadria, Liam Roditty, and Virginia Vassilevska Williams

Published in: LIPIcs, Volume 388, 34th Annual European Symposium on Algorithms (ESA 2026)

Abstract
We study the problem of approximating the length of a shortest cycle in a given graph, known as the girth of the graph. The state-of-the-art approximation algorithms for unweighted graphs by Kadria et al. [SODA'22] and Roditty and Trabelsi [arXiv'25] achieve the following trade-off: for every integer k ≥ 2, there is an Õ(n^{1+2/k}) time algorithm that achieves a (2k/3)-approximation for the girth in unweighted n-node graphs. The first result of this paper is to achieve the same trade-off for m-edge, n-node graphs with non-negative real edge weights: a 2k/3-approximation algorithm running in Õ(m+n^{1+2/k}) time. The dependence on m is unavoidable in weighted graphs. Our result improves on the work of Kadria et al. [SODA'23] and Ducoffe [ICALP'19 and SIDMA'21], who were only able to achieve such a trade-off for some values of k. We also prove new fine-grained lower bounds for girth approximation and related problems in unweighted graphs.
Cite as

Avi Kadria, Liam Roditty, and Virginia Vassilevska Williams. Tighter Bounds for Weighted and Unweighted Shortest Cycle Approximation. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 82:1-82:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kadria_et_al:LIPIcs.ESA.2026.82,
  author =	{Kadria, Avi and Roditty, Liam and Vassilevska Williams, Virginia},
  title =	{{Tighter Bounds for Weighted and Unweighted Shortest Cycle Approximation}},
  booktitle =	{34th Annual European Symposium on Algorithms (ESA 2026)},
  pages =	{82:1--82:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-445-1},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{388},
  editor =	{Bille, Philip and Pettie, Seth and Storandt, Sabine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.82},
  URN =		{urn:nbn:de:0030-drops-272180},
  doi =		{10.4230/LIPIcs.ESA.2026.82},
  annote =	{Keywords: Fine-grained complexity, Graph algorithms, shortest cycle, girth approximations}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2026.121
Faster Algorithms for (2k-1)-Stretch Distance Oracles

Authors: Avi Kadria and Liam Roditty

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract
The seminal distance oracles of Thorup and Zwick [STOC 2001, JACM 2005] provide optimal stretch/space tradeoffs. However, their O(mn^{1/k}) construction time is not optimal, and they posed the question of whether a faster construction time is possible (especially for small k). In this paper, we present the first improvement upon their construction algorithm in graphs that are not super sparse, i.e., when m = Ω(n^{1+1/k+ε}), for any ε > 0. Moreover, our construction improves upon the O(n²)-time construction of Baswana and Kavitha [FOCS 2006, SICOMP 2010], for every k > 2. By achieving the first subquadratic construction for 2 < k < 6, we resolve the open problem posed by Wulff-Nilsen [SODA 2012] of whether such subquadratic-time constructions exist. Wulff-Nilsen [SODA 2012] targeted nearly linear construction times and presented algorithms running in Õ(m + n^{1+f(k)}) time, which is near-linear whenever the graph density m exceeds the threshold n^{1+f(k)}. We obtain improved bounds on f(k) for all k > 3, and thus expand the regime of graph densities for which nearly linear construction times are achievable. In addition, for unweighted graphs, we present several new algorithms for constructing (2k - 1,β)-oracles that improve upon the results of Baswana, Gaur, Sen, and Upadhyay [ICALP 2008]. Our results are achieved through the development of several new algorithmic tools, which may be of independent interest. One of our main technical contributions is a hierarchy of parameterized distance oracles, which plays a central role in our fast construction algorithms.
Cite as

Avi Kadria and Liam Roditty. Faster Algorithms for (2k-1)-Stretch Distance Oracles. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 121:1-121:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kadria_et_al:LIPIcs.ICALP.2026.121,
  author =	{Kadria, Avi and Roditty, Liam},
  title =	{{Faster Algorithms for (2k-1)-Stretch Distance Oracles}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{121:1--121:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael 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.2026.121},
  URN =		{urn:nbn:de:0030-drops-265105},
  doi =		{10.4230/LIPIcs.ICALP.2026.121},
  annote =	{Keywords: Fine-grained complexity, Graph algorithms, shortest cycle, girth approximations}
}
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.2025.38
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)

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@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}
}

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