Finding single-source shortest p-disjoint paths: fast computation and sparse preservers

@inproceedings{Bil2022FindingSS,
  title={Finding single-source shortest p-disjoint paths: fast computation and sparse preservers},
  author={Davide Bil{\`o} and Gianlorenzo D'angelo and Luciano Gual{\`a} and Stefano Leucci and Guido Proietti and Mirko Rossi},
  booktitle={STACS},
  year={2022}
}
Let G be a directed graph with n vertices, m edges, and non-negative edge costs. Given G, a fixed source vertex s, and a positive integer p, we consider the problem of computing, for each vertex t ̸= s, p edge-disjoint paths of minimum total cost from s to t in G. Suurballe and Tarjan [Networks, 1984] solved the above problem for p = 2 by designing a O(m + n log n) time algorithm which also computes a sparse single-source 2-multipath preserver, i.e., a subgraph containing 2 edge-disjoint paths… 

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References

SHOWING 1-10 OF 31 REFERENCES

A quick method for finding shortest pairs of disjoint paths

TLDR
This paper considers the problem of finding, for each possible sink vertex v, a pair of edge-disjoint paths from s to v of minimum total edge cost, and gives an implementation of Suurballe's algorithm that runs in O(m log(1+ m/n)n) time and O( m) space.

Approximate Single Source Fault Tolerant Shortest Path

TLDR
This paper addresses several variants of the problem of maintaining the (1 + ∈)-approximate shortest path from s to each v ∈ V in the presence of a failure of an edge or a vertex and shows that the size of the subgraph H is optimal (up to logarithmic factors) by proving a lower bound of Ω(nL/∈) edges.

Node-Disjoint Multipath Spanners and Their Relationship with Fault-Tolerant Spanners

TLDR
Building upon recent results on fault-tolerant spanners, it is shown how to build p-multipath spanners of constant stretch and of ${\tilde{O}}(n^{1+1/k})$ edges, for fixed parameters p and k, n being the number of nodes of the graph.

Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees

TLDR
The problem of designing a sparse f -edge-fault-tolerant ( f -EFT) σ -approximate single-source shortest-path tree (ASPT) is studied, namely a subgraph of G having as few edges as possible and which contains paths from a fixed source that are stretched by a factor of at most σ .

Multipath Spanners via Fault-Tolerant Spanners

TLDR
It is shown that at the cost of increasing the number of edges by a polynomial factor in p and s, it is possible to obtain an s-multipath spanner, thereby improving on the large stretch obtained in [15,16].

Sparse Fault-Tolerant BFS Structures

TLDR
This article considers breadth-first search (BFS) spanning trees and addresses the problem of designing a sparse fault-tolerant BFS structure (FT-BFS structure), namely, a sparse subgraph T of the given network G such that subsequent to the failure of a single edge or vertex, the surviving part T′ of T still contains a BFS spanning tree for (the surviving part of) G.

Multipath Spanners

TLDR
Spanners that approximate short cycles, and more generally p edge-disjoint paths with p>1, between any pair of vertices are studied.

Better Distance Preservers and Additive Spanners

TLDR
It is proved that consistency alone is not enough to understand distance preservers, by showing both a lower bound on the power of consistency and a new general upper bound that polynomially surpasses it.

Fault Tolerant Approximate BFS Structures

TLDR
This paper addresses the problem of designing a fault-tolerant (α, β) approximate BFS structure (or FT-ABFSstructure for short), namely, a subgraph H of the network G such that subsequent to the failure of some subset F of edges or vertices, the surviving part of H still contains an approximate B FS spanning tree for (the surviving parts of) G.

A matroid approach to finding edge connectivity and packing arborescences

  • H. Gabow
  • Computer Science, Mathematics
    STOC '91
  • 1991
TLDR
An algorithm that finds k edge-disjoint arborescences on a directed graph in time O(kmn + k3n2)2 is presented, based on two theorems of Edmonds that link these two problems and show how they can be solved.