TY - GEN
T1 - Faster Algorithms for Dual-Failure Replacement Paths
AU - Chechik, Shiri
AU - Zhang, Tianyi
N1 - Publisher Copyright: © Shiri Chechik and Tianyi Zhang.
PY - 2024/7
Y1 - 2024/7
N2 - Given a simple weighted directed graph G = (V, E, ω) on n vertices as well as two designated terminals s, t ∈ V , our goal is to compute the shortest path from s to t avoiding any pair of presumably failed edges f1, f2 ∈ E, which is a natural generalization of the classical replacement path problem which considers single edge failures only. This dual failure replacement paths problem was recently studied by Vassilevska Williams, Woldeghebriel and Xu [FOCS 2022] who designed a cubic time algorithm for general weighted digraphs which is conditionally optimal; in the same paper, for unweighted graphs where ω ≡ 1, the authors presented an algebraic algorithm with runtime Õ(n2.9146), as well as a conditional lower bound of n8/3−o(1) against combinatorial algorithms. However, it was unknown in their work whether fast matrix multiplication is necessary for a subcubic runtime in unweighted digraphs. As our primary result, we present the first truly subcubic combinatorial algorithm for dual failure replacement paths in unweighted digraphs. Our runtime is Õ(n3−1/18). Besides, we also study algebraic algorithms for digraphs with small integer edge weights from {−M, −M +1, · · ·, M −1, M}. As our secondary result, we obtained a runtime of Õ(Mn2.8716), which is faster than the previous bound of Õ(M2/3n2.9144 + Mn2.8716) from [Vassilevska Williams, Woldeghebriela and Xu, 2022].
AB - Given a simple weighted directed graph G = (V, E, ω) on n vertices as well as two designated terminals s, t ∈ V , our goal is to compute the shortest path from s to t avoiding any pair of presumably failed edges f1, f2 ∈ E, which is a natural generalization of the classical replacement path problem which considers single edge failures only. This dual failure replacement paths problem was recently studied by Vassilevska Williams, Woldeghebriel and Xu [FOCS 2022] who designed a cubic time algorithm for general weighted digraphs which is conditionally optimal; in the same paper, for unweighted graphs where ω ≡ 1, the authors presented an algebraic algorithm with runtime Õ(n2.9146), as well as a conditional lower bound of n8/3−o(1) against combinatorial algorithms. However, it was unknown in their work whether fast matrix multiplication is necessary for a subcubic runtime in unweighted digraphs. As our primary result, we present the first truly subcubic combinatorial algorithm for dual failure replacement paths in unweighted digraphs. Our runtime is Õ(n3−1/18). Besides, we also study algebraic algorithms for digraphs with small integer edge weights from {−M, −M +1, · · ·, M −1, M}. As our secondary result, we obtained a runtime of Õ(Mn2.8716), which is faster than the previous bound of Õ(M2/3n2.9144 + Mn2.8716) from [Vassilevska Williams, Woldeghebriela and Xu, 2022].
KW - graph algorithms
KW - replacement paths
KW - shortest paths
UR - http://www.scopus.com/inward/record.url?scp=85198353517&partnerID=8YFLogxK
U2 - https://doi.org/10.4230/LIPIcs.ICALP.2024.41
DO - https://doi.org/10.4230/LIPIcs.ICALP.2024.41
M3 - منشور من مؤتمر
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024
A2 - Bringmann, Karl
A2 - Grohe, Martin
A2 - Puppis, Gabriele
A2 - Svensson, Ola
T2 - 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024
Y2 - 8 July 2024 through 12 July 2024
ER -