Keywords and Synonyms
Incremental algorithms for graphs; Fully dynamic graph algorithm for maintaining connectivity
Problem Definition
Design a data structure for an undirected graph with a fixed set of nodes which can process queries of the form “Are nodes i and j connected?” and updates of the form “Insert edge \( { \{i,j\} } \)”; “Delete edge \( { \{i,j\} } \).” The goal is to minimize update and query times, over the worst-case sequence of queries and updates. Algorithms to solve this problem are called “fully dynamic” as opposed to “partially dynamic” since both insertions and deletions are allowed.
Key Results
Holm et al. [4] gave the first deterministic fully dynamic graph algorithm for maintaining connectivity in an undirected graph with polylogarithmic amortized time per operation, specifically, \( { O(\log^2 n) } \) amortized cost per update operation and \( { O(\log n/\log \log n) } \) worst-case per query, where nis the number of nodes. The basic technique is extended to...
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Recommended Reading
Eppstein, D., Galil, Z., Italiano, G.F., Nissenzweig, A.:. Sparsification–a technique for speeding up dynamic graph algorithms. J. ACM 44(5), 669–696.1 (1997)
Henzinger, M.R., King, V.: Randomized fully dynamic graph algorithms with polylogarithmic time per operation. J. ACM 46(4), 502–536 (1999) (presented at ACM STOC 1995)
Henzinger, M.R., Thorup, M.: Sampling to provide or to bound: With applications to fully dynamic graph algorithms. Random Struct. Algorithms 11(4), 369–379 (1997) (presented at ICALP 1996)
Holm, J., De Lichtenberg, K., Thorup, M.: Poly-logarithmic Deterministic Fully-Dynamic Algorithms for Connectivity, Minimum Spanning Tree, 2-Edge, and Biconnectivity. J. ACM 48(4), 723–760 (2001) (presented at ACM STOC 1998)
Iyer, R., Karger, D., Rahul, H., Thorup, M.: An experimental study of poly-logarithmic fully-dynamic connectivity algorithms. J. Exp. Algorithmics 6(4) (2001) (presented at ALENEX 2000)
Pătraşcu, M., Demaine, E.: Logarithmic Lower Bounds in the Cell-Probe Model. SIAM J. Comput. 35(4), 932–963 (2006) (presented at ACM STOC 2004)
Thorup, M.: Near-optimal fully-dynamic graph connectivity. In: Proceedings of the 32th ACM Symposium on Theory of Computing pp. 343–350. ACM STOC (2000)
Thorup, M.: Dynamic Graph Algorithms with Applications. In: Halldórsson, M.M. (ed) 7th Scandinavian Workshop on Algorithm Theory (SWAT), Norway, 5–7 July 2000, pp. 1–9
Zaroliagis, C.D.: Implementations and experimental studies of dynamic graph algorithms. In: Experimental Algorithmics, Dagstuhl seminar, September 2000, Lecture Notes in Computer Science, vol. 2547. Springer (2002), Journal Article: J. Exp. Algorithmics 229–278 (2000)
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King, V. (2008). Fully Dynamic Connectivity. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_152
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DOI: https://doi.org/10.1007/978-0-387-30162-4_152
Publisher Name: Springer, Boston, MA
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