Min-Cuts and Shortest Cycles in Planar Graphs in O(n loglogn) Time

Abstract

We present a deterministic O(n log log n) time algorithm for finding shortest cycles and minimum cuts in planar graphs. The algorithm improves the previously known fastest algorithm by Italiano et al. in STOC'11 by a factor of log n. This speedup is obtained through the use of dense distance graphs combined with a divide-and-conquer approach.

Global Minimum Cuts in Surface Embedded Graphs

“…1 Chalermsook et al [13] described the first algorithm to compute global minimum cuts in undirected planar graphs in near-linear time. Their algorithm was recently improved by Łącki and Sankowski [53] to run in O(n log log n) time.…”

“…Let γ 1 and γ 2 be the shortest simple cycles in the subgraphs G Q \e 1 and G Q \e 2 , respectively. We compute both γ 1 and γ 2 in O(g n log log n) time using the algorithm of Łącki and Sankowski [53].…”

“…Our algorithm repeatedly applies two recent O(n log log n)-time algorithms for planar graphs as black boxes: one due to Łącki and Sankowski for global minimum cuts [53] and another due to Italiano et al for minimum (s, t)-cuts [46]. Indeed, these are the only subroutines in our algorithm that require more than linear time when the genus is fixed.…”

“…Chalermsook, Fakcharoenphol, and Nanongkai described the first deterministic near-linear time algorithm for computing minimum cuts in planar graphs [13]. This result was recently improved by Italiano et al [46] and again by Łącki and Sankowski [53], who describe a deterministic algorithm that runs in O(n log log n) time. These algorithms are significantly faster than the best known algorithms for general sparse graphs: a randomized algorithm of Karger that runs in O(n log 3 n) time and succeeds with high probability [49], and a deterministic algorithm of Nagamochi and Ibaraki that runs in O(n 2 log n) time [60].…”

Minimum Cut of Directed Planar Graphs in O(n log log n) Time

“…Before each recursive level we can remove every vertex of degree two, and merge its two adjacent edges into a single edge (combining the lengths of the two). This guarantees that the total size of all subgraphs in the same level of the recursion is O(n) [13,21], and so all executions of the min st-cut algorithm in Copyright © 2018 by SIAM Unauthorized reproduction of this article is prohibited 480 Downloaded 05/11/18 to 34.210.69.67. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php the conquering steps at this level take total O(n log n).…”