Abstract
We present a deterministic incremental algorithm for exactly maintaining the size of a minimum cut with O (log 3 n log log 2 n ) amortized time per edge insertion and O (1) query time. This result partially answers an open question posed by Thorup (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 sparsification technique of Kawarabayashi and Thorup (2015) or its recent improvement by Henzinger, Rao, and Wang (2017), and an exact incremental algorithm of Henzinger (1997). We also study space-efficient incremental algorithms for the minimum cut problem. Concretely, we show that there exists an O ( n log n /ε 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+ε)-approximation to the minimum cut. The algorithm has O ((α ( n ) log 3 n )/ε 2 ) amortized update time and constant query time, where α ( n ) stands for the inverse of Ackermann function.
Highlights
Computing a minimum cut of a graph is a fundamental algorithmic graph problem
We present two new incremental algorithms concerning the maintenance of the size of a minimum cut
This result allows us to partially answer in the affirmative a question regarding efficient dynamic algorithms for exact minimum cut posed by Thorup [27]
Summary
Computing a minimum cut of a graph is a fundamental algorithmic graph problem. 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. 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. 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. 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 insertions starting from an empty graph.
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