Affiliation:
1. University of Vienna, Austria
2. University of Copenhagen, Copenhagen East, Denmark
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.
Funder
Danish Council for Independent Research under the Sapere Aude research career programme
European Research Council under the European Union’s 7th Framework Programme
ERC
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
8 articles.
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