Incremental Exact Min-Cut in Poly-logarithmic Amortized Update Time

Incremental Exact Min-Cut in Poly-logarithmic Amortized Update Time∗ Gramoz Goranci1 , Monika Henzinger†2 , and Mikkel Thorup‡3 1 2 3 University of Vienna, Faculty of Computer Science, Vienna, Austria University of Vienna, Faculty of Computer Science, Vienna, Austria Faculty of Computer Science, University of Copenhagen, Copenhagen, Denmark Abstract We present a deterministic incremental algorithm for exactly maintaining the size of a minimum e 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/ε2 ) space Monte-Carlo algorithm that can process a stream of edge insertions starting from an empty graph, and with high probability, the e algorithm maintains a (1 + ε)-approximation to the minimum cut. The algorithm has O(1) amortized update-time and constant query-time. 1998 ACM Subject Classification G.2.2 Graph Theory Keywords and phrases Dynamic Graph Algorithms, Minimum Cut, Edge Connectivity Digital Object Identifier 10.4230/LIPIcs.ESA.2016.46 1 Introduction Computing a minimum cut of a graph is a fundamental algorithmic graph problem. While most of the focus has been on designing static efficient algorithms for finding a minimum cut, the dynamic maintenance of a minimum cut has also attracted increasing attention over the last two decades. The motivation for studying the dynamic setting is apparent, as important networks such as social or road network undergo constant and rapid changes. Given an initial graph G, the goal of a dynamic graph algorithm is to build a datastructure that maintains G and supports update and query operations. Depending on the types of update operations we allow, dynamic algorithms are classified into three main ∗ This work was done in part while M. Henzinger and M. Thorup were visiting the Simons Institute for the Theory of Computing. † The research leading to these results has received funding from the European Research Council under the European Union’s 7th Framework Programme (FP/2007-2013) / ERC Grant Agreement no. 340506 for M. Henzinger. ‡ M. Thorup’s research is partly supported by Advanced Grant DFF-0602-02499B from the Danish Council for Independent Research under the Sapere Aude research career programme. © Gramoz Goranci, Monika Henzinger, and Mikkel Thorup; licensed under Creative Commons License CC-BY 24th Annual European Symposium on Algorithms (ESA 2016). Editors: Piotr Sankowski and Christos Zaroliagis; Article No. 46; pp. 46:1–46:17 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 46:2 Incremental Exact Min-Cut in Poly-logarithmic Amortized Update Time categories: (i) fully dynamic, if update operations consist of both edge insertions and deletions, (ii) incremental, if update operations consist of edge insertions only and (iii) decremental, if update operations consist of edge deletions only. In this paper, we study incremental algorithms for maintaining the size of a minimum cut of an unweighted, undirected graph (denoted by λ(G) = λ) supporting the following operations: Insert(u, v): insert the edge (u, v) in G. QuerySize: return the size of a minimum cut of the current G. For any α ≥ 1, we say that an algorithm is an α-approximation of λ if QuerySize returns a positive number k such that λ ≤ k ≤ α · λ. Our problem is characterized by two time measures; query time, which denotes the time needed to answer each query and total update time, which denotes the time needed to process all edge insertions. We say that an algorithm has an O(t(n)) amortized update time if it takes O(m(t(n))) total update time for m edge e to hide poly-logarithmic factors. insertions starting from an empty graph. We use O(·) Related Work For over a decade, the best known static and deterministic algorithm for computing a minimum cut was due to Gabow [10] which runs in O(m + λ2 log n) time. Recently, Kawarabayashi and e Thorup [19] devised a O(m) time algorithm which applies only to simple, unweighted and undirected graphs. Randomized Monte Carlo algorithms in the context of static minimum cut were initiated by Karger [17]. The best known randomized algorithm is due to Karger [18] and runs in O(m log3 n) time. Karger [16] was the first to study the dynamic maintenance of a minimum cut in its full algorithm for maintaining pgenerality. He devised a fully dynamic, albeit randomized, 1/2+ε e a 1 + 2/ε-approximation of the minimum cut in O(n ) expected amortized time per edge operation. In the incremental setting, he showed that the update time for the same e ε ). Thorup and Karger [28] improved approximation ratio can be further improved to O(n p e upon the above guarantees by achieving an approximation factor of 2 + o(1) and an O(1) expected amortized time per edge operation. Henzinger [14] obtained the following guarantees for the incremental minimum cut; for any ε ∈ (0, 1], (i) an O(1/ε2 ) amortized update-time for a (2 + ε)-approximation, (ii) an O(log3 n/ε2 ) expected amortized update-time for a (1 + ε)-approximation and (iii) an O(λ log n) amortized update-time for the exact minimum cut. For minimum cut up to some poly-logarithmic size, Thorup [27] gave a fully dynamic e √n) time per edge operation. Monte-Carlo algorithm for maintaining exact minimum cut in O( He also showed how to obtain an 1 + o(1)-approximation of an arbitrary sized minimum cut with the same time bounds. In comparison to previous results, it is worth pointing out that his work achieves worst-case update times. Łącki and Sankwoski [21] studied the dynamic maintenance of the exact size of the minimum cut in planar graphs with arbitrary edge weights. They obtained a fully dynamic e 5/6 ) query and update time. algorithm with O(n There has been a growing interest in proving conditional lower bounds for dynamic problems in the last few years [1, 13]. A recent result of Nanongkai and Saranurak [24] shows the following conditional lower-bound for the exact weighted minimum cut assuming the Online Matrix-Vector Multiplication conjecture: for any ε > 0, there are no fully-dynamic algorithms with polynomial-time preprocessing that can simultaneously achieve O(n1−ε ) update-time and O(n2−ε ) query-time. G.Goranci, M. Henzinger, and M. Thorup 46:3 Results and Technical Overview We present two new incremental algorithms concerning the maintenance of the size of a minimum cut. Both algorithms apply to undirected, unweighted simple graphs. Our first and main resu (...truncated)