We present a deterministic incremental algorithm for exactly maintaining the size of a minimum cut with ~O(1) amortized time per edge insertion and O(1) query time. This result partially answers an open question posed by Thorup [Combinatorica 2007]. It also stays in sharp contrast to a polynomial conditional lower-bound for the fully-dynamic weighted minimum cut problem. Our algorithm is obtained by combining a recent sparsification technique of Kawarabayashi and Thorup [STOC 2015] and an exact incremental algorithm of Henzinger [J. of Algorithm 1997]. We also study space-efficient incremental algorithms for the minimum cut problem. Concretely, we show that there exists an O(n log n/epsilon^2) space Monte-Carlo algorithm that can process a stream of edge insertions starting from an empty graph, and with high probability, the algorithm maintains a (1+epsilon)-approximation to the minimum cut. The algorithm has ~O(1) amortized update-time and constant query-time.
@InProceedings{goranci_et_al:LIPIcs.ESA.2016.46, author = {Goranci, Gramoz and Henzinger, Monika and Thorup, Mikkel}, title = {{Incremental Exact Min-Cut in Poly-logarithmic Amortized Update Time}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {46:1--46:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.46}, URN = {urn:nbn:de:0030-drops-63584}, doi = {10.4230/LIPIcs.ESA.2016.46}, annote = {Keywords: Dynamic Graph Algorithms, Minimum Cut, Edge Connectivity} }
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