Decremental All-Pairs ALL Shortest Paths and Betweenness Centrality

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8889)


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 through any given vertex. Our APASP algorithm can be used for the decremental computation of betweenness centrality (BC), which is widely used in the analysis of large complex networks. No nontrivial decremental algorithm for either problem was known prior to our work. Our method is a generalization of the decremental algorithm of Demetrescu and Italiano [3] for unique shortest paths, and for graphs with \({\nu ^*}= O(n)\), we match the bound in [3]. Thus for graphs with a constant number of shortest paths between any pair of vertices, our algorithm maintains APASP and BC scores in amortized time \(O(n^2 \cdot \log n)\) under decremental updates, regardless of the number of edges in the graph.


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  1. 1.
    Brandes, U.: A faster algorithm for betweenness centrality. Journal of Mathematical Sociology 25(2), 163–177 (2001)CrossRefMATHGoogle Scholar
  2. 2.
    Coffman, T., Greenblatt, S., Marcus, S.: Graph-based technologies for intelligence analysis. Commun. ACM 47(3), 45–47 (2004)CrossRefGoogle Scholar
  3. 3.
    Demetrescu, C., Italiano, G.F.: A new approach to dynamic all pairs shortest paths. J. ACM 51(6), 968–992 (2004)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Freeman, L.C.: A set of measures of centrality based on betweenness. Sociometry 40(1), 35–41 (1977)CrossRefGoogle Scholar
  5. 5.
    Green, O., McColl, R., Bader, D.A.: A fast algorithm for streaming betweenness centrality. In: Proc. of 4th PASSAT, pp. 11–20 (2012)Google Scholar
  6. 6.
    Karger, D.R., Koller, D., Phillips, S.J.: Finding the hidden path: Time bounds for all-pairs shortest paths. SIAM J. Comput. 22(6), 1199–1217 (1993)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Kourtellis, N., Alahakoon, T., Simha, R., Iamnitchi, A., Tripathi, R.: Identifying high betweenness centrality nodes in large social networks. In: Social Network Analysis and Mining, pp. 1–16 (2012)Google Scholar
  8. 8.
    Krebs, V.: Mapping networks of terrorist cells. CONNECTIONS 24(3), 43–52 (2002)Google Scholar
  9. 9.
    Lee, M.-J., Lee, J., Park, J.Y., Choi, R.H., Chung, C.-W.: Qube: a quick algorithm for updating betweenness centrality. In: Proc. of 21st WWW, pp. 351–360 (2012)Google Scholar
  10. 10.
    Nasre, M., Pontecorvi, M., Ramachandran, V.: Betweenness Centrality – Incremental and Faster. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014, Part II. LNCS, vol. 8635, pp. 577–588. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  11. 11.
    Pinney, J.W., McConkey, G.A., Westhead, D.R.: Decomposition of biological networks using betweenness centrality. In: Proc. of RECOMB (2005)Google Scholar
  12. 12.
    Pontecorvi, M., Ramachandran, V.: Fully dynamic all pairs all shortest paths and betweenness centrality. (2014) (manuscript)Google Scholar
  13. 13.
    Goel, K., Singh, R.R., Iyengar, S., Sukrit, : A Faster Algorithm to Update Betweenness Centrality after Node Alteration. In: Bonato, A., Mitzenmacher, M., Prałat, P. (eds.) WAW 2013. LNCS, vol. 8305, pp. 170–184. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  14. 14.
    Thorup, M.: Fully-Dynamic All-Pairs Shortest Paths: Faster and Allowing Negative Cycles. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 384–396. Springer, Heidelberg (2004)CrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Indian Institute of TechnologyMadrasIndia
  2. 2.University of Texas at AustinAustinUSA

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