Decremental All-Pairs ALL Shortest Paths and Betweenness Centrality

@article{Nasre2014DecrementalAA,
  title={Decremental All-Pairs ALL Shortest Paths and Betweenness Centrality},
  author={Meghana Nasre and Matteo Pontecorvi and Vijaya Ramachandran},
  journal={ArXiv},
  year={2014},
  volume={abs/1411.4073}
}
We consider the all pairs all shortest paths (APASP) problem, which maintains the shortest path dag rooted at every vertex in a directed graph \(G=(V,E)\) with positive edge weights. For this problem we present a decremental algorithm (that supports the deletion of a vertex, or weight increases on edges incident to a vertex). Our algorithm runs in amortized \(O({\nu ^*}^2 \cdot \log n)\) time per update, where \(n = |V| \), and \({\nu ^*}\) bounds the number of edges that lie on shortest paths… 
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This work presents a fully dynamic algorithm for the all pairs all shortest paths (APASP) problem, which maintains all of the multiple shortest paths for every vertex pair in a directed graph G=(V,E) with a positive real weight on each edge.
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We present an incremental algorithm that updates the betweenness centrality (BC) score of all vertices in a graph G when a new edge is added to G, or the weight of an existing edge is reduced. Our
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