research-article

Replacement Paths and Distance Sensitivity Oracles via Fast Matrix Multiplication

Authors:

Raphael YusterAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 9, Issue 2

Article No.: 14, Pages 1 - 13

https://doi.org/10.1145/2438645.2438646

Published: 01 March 2013 Publication History

Abstract

A distance sensitivity oracle of an n-vertex graph G = (V,E) is a data structure that can report shortest paths when edges of the graph fail. A query (u ∈ V, v ∈ V, S ⊆ E) to this oracle returns a shortest u-to-v path in the graph G^′ = (V,E ∖ S). We present randomized (Monte Carlo) algorithms for constructing a distance sensitivity oracle of size Õ(n^3−α) for |S| = O(lg n/lg lg n) and any choice of 0 < α < 1. For real edge-lengths, the oracle is constructed in O(n^4−α) time and a query to this oracle takes Õ(n^{2−2(1−α)/|S|}) time. For integral edge-lengths in {−M,..., M}, using the current ω < 2.376 matrix multiplication exponent, the oracle is constructed in O(Mn^3.376−α) time with Õ(n^{2−(1−α)/|S|}) query, or alternatively in O(M^0.681n^3.575−α) time with Õ(n^{2−2(1−α)/|S|}) query.

Distance sensitivity oracles generalize the replacement paths problem in which u and v are known in advance and |S| = 1. In other words, if P is a shortest path from u to v in G, then the replacement paths problem asks to compute, for every edge e on P, a shortest u-to-v path that avoids e. Our new technique for constructing distance sensitivity oracles using fast matrix multiplication also yields the first subcubic-time algorithm for the replacement paths problem when the edge-lengths are small integers. In particular, it yields a randomized (Monte Carlo) Õ(Mn^2.376 + M^{2 3} n^2.584)-time algorithm for the replacement paths problem assuming M ≤ n^0.624.

Finally, we mention that both our replacement paths algorithm and our distance sensitivity oracle can be made to work, in the same time and space bounds, for the case of failed vertices rather than edges, that is, when S is a set of vertices and we seek a shortest u-to-v path in the graph obtained from G by removing all vertices in S and their adjacent edges.

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Cited By

C. S. KParter M(2024)Deterministic Replacement Path CoveringACM Transactions on Algorithms10.1145/367376020:4(1-35)Online publication date: 5-Aug-2024
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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 9, Issue 2

March 2013

89 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/2438645

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Copyright © 2013 ACM.

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Publication History

Published: 01 March 2013

Accepted: 01 January 2012

Revised: 01 December 2011

Received: 01 February 2011

Published in TALG Volume 9, Issue 2

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Cited By

C. S. KParter M(2024)Deterministic Replacement Path CoveringACM Transactions on Algorithms10.1145/367376020:4(1-35)Online publication date: 5-Aug-2024
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Parter MPetruschka APettie SMohar BShinkar IO'Donnell R(2024)Connectivity Labeling and Routing with Multiple Vertex FailuresProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649729(823-834)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649729
Dey DGupta MMohar BShinkar IO'Donnell R(2024)Nearly Optimal Fault Tolerant Distance OracleProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649697(944-955)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649697
Bilò DChechik SChoudhary KCohen SFriedrich TSchirneck M(2024)Improved Distance (Sensitivity) Oracles with Subquadratic Space2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00097(1550-1558)Online publication date: 27-Oct-2024
https://doi.org/10.1109/FOCS61266.2024.00097
Parter MPetruschka A(2024)Near-optimal distributed computation of small vertex cutsDistributed Computing10.1007/s00446-023-00455-z37:2(67-88)Online publication date: 1-Jun-2024
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Bodwin GParter M(2023)Restorable Shortest Path Tiebreaking for Edge-Faulty GraphsJournal of the ACM10.1145/360354270:5(1-24)Online publication date: 11-Oct-2023
https://dl.acm.org/doi/10.1145/3603542
Bilò DChechik SChoudhary KCohen SFriedrich TKrogmann SSchirneck MSaha BServedio R(2023)Approximate Distance Sensitivity Oracles in Subquadratic SpaceProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585251(1396-1409)Online publication date: 2-Jun-2023
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Abboud AGrandoni FVassilevska Williams V(2023)Subcubic Equivalences between Graph Centrality Problems, APSP, and DiameterACM Transactions on Algorithms10.1145/356339319:1(1-30)Online publication date: 9-Mar-2023
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Karczmarz ASankowski P(2023)Sensitivity and Dynamic Distance Oracles via Generic Matrices and Frobenius Form2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00106(1745-1756)Online publication date: 6-Nov-2023
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Bilò DChoudhary KCohen SFriedrich TKrogmann SSchirneck M(2023)Compact Distance Oracles with Large Sensitivity and Low StretchAlgorithms and Data Structures10.1007/978-3-031-38906-1_11(149-163)Online publication date: 28-Jul-2023
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