Minimum Cuts and Shortest Cycles in Directed Planar Graphs via Noncrossing Shortest Paths
Abstract
Let $G$ be an $n$-node simple directed planar graph with nonnegative edge weights. We study the fundamental problems of computing (1) a global cut of $G$ with minimum weight and (2) a cycle of $G$ with minimum weight. The best previously known algorithm for the former problem, running in $O(n\log^3 n)$ time, can be obtained from the algorithm of Ła̧cki, Nussbaum, Sankowski, and Wulff-Nilsen for single-source all-sinks maximum flows. The best previously known result for the latter problem is the $O(n\log^3 n)$-time algorithm of Wulff-Nilsen. By exploiting duality between the two problems in planar graphs, we solve both problems in $O(n\log n\log\log n)$ time via a divide-and-conquer algorithm that finds a shortest nondegenerate cycle. The kernel of our result is an $O(n\log\log n)$-time algorithm for computing noncrossing shortest paths among nodes well ordered on a common face of a directed plane graph, which is extended from the algorithm of Italiano, Nussbaum, Sankowski, and Wulff-Nilsen for an undirected plane graph.
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References
1.
G. Borradaile and P. N. Klein, An $O(n\log n)$ algorithm for maximum $st$-flow in a directed planar graph, J. ACM, 56 (2009), pp. 9.1--9.30.
2.
G. Borradaile, P. N. Klein, S. Mozes, Y. Nussbaum, and C. Wulff-Nilsen, Multiple-source multiple-sink maximum flow in directed planar graphs in near-linear time, in Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2011, pp. 170--179.
3.
G. Borradaile, P. Sankowski, and C. Wulff-Nilsen, Min $st$-cut oracle for planar graphs with near-linear preprocessing time, ACM Trans. Algorithms, 11 (2015), pp. 16.
4.
U. Brandes and D. Wagner, A linear time algorithm for the arc disjoint Menger problem in planar directed graphs, Algorithmica, 28 (2000), pp. 16--36.
5.
S. Cabello, Finding shortest contractible and shortest separating cycles in embedded graphs, ACM Trans. Algorithms, 6 (2010), pp. 24.
6.
S. Cabello, E. W. Chambers, and J. Erickson, Multiple-source shortest paths in embedded graphs, SIAM J. Comput., 42 (2013), pp. 1542--1571.
7.
S. Cabello, É. Colin de Verdière, and F. Lazarus, Finding shortest non-trivial cycles in directed graphs on surfaces, in Proceedings of the 26th ACM Symposium on Computational Geometry, ACM, New York, 2010, pp. 156--165.
8.
P. Chalermsook, J. Fakcharoenphol, and D. Nanongkai, A deterministic near-linear time algorithm for finding minimum cuts in planar graphs, in Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2004, pp. 828--829.
9.
H.-C. Chang and H.-I. Lu, Computing the girth of a planar graph in linear time, SIAM J. Comput., 42 (2013), pp. 1077--1094.
10.
T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 3rd ed., MIT Press, Cambridge, MA, 2009.
11.
M. Cygan, H. N. Gabow, and P. Sankowski, Algorithmic applications of Baur-Strassen's theorem: Shortest cycles, diameter and matchings, in Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2012, pp. 531--540.
12.
H. Djidjev, A faster algorithm for computing the girth of planar and bounded genus graphs, ACM Trans. Algorithms, 7 (2010), pp. 3.
13.
D. Eisenstat and P. N. Klein, Linear-time algorithms for max flow and multiple-source shortest paths in unit-weight planar graphs, in Proceedings of the 45th ACM Symposium on Theory of Computing, ACM, New York, 2013, pp. 735--744.
14.
J. Erickson, Maximum flows and parametric shortest paths in planar graphs, in Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2010, pp. 794--804.
15.
J. Erickson, K. Fox, and A. Nayyeri, Global minimum cuts in surface embedded graphs, in Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2012, pp. 1309--1318.
16.
J. Erickson and S. Har-Peled, Optimally cutting a surface into a disk, Discrete Comput. Geom., 31 (2004), pp. 37--59.
17.
J. Erickson and A. Nayyeri, Minimum cuts and shortest non-separating cycles via homology covers, in Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2011, pp. 1166--1176.
18.
J. Erickson and A. Nayyeri, Shortest non-crossing walks in the plane, in Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2011, pp. 297--208.
19.
J. Erickson and P. Worah, Computing the shortest essential cycle, Discrete Comput. Geom., 44 (2010), pp. 912--930.
20.
J. Fakcharoenphol and S. Rao, Planar graphs, negative weight edges, shortest paths, and near linear time, J. Comput. System Sci., 72 (2006), pp. 868--889.
21.
K. Fox, Shortest non-trivial cycles in directed and undirected surface graphs, in Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2013, pp. 352--364.
22.
K. Fox, Fast Algorithms for Surface Embedded Graphs via Homology, PhD thesis, University of Illinois at Urbana-Champaign, Champaign, IL, 2014.
23.
G. N. Frederickson, Fast algorithms for shortest paths in planar graphs, with applications, SIAM J. Comput., 16 (1987), pp. 1004--1022.
24.
H. N. Gabow, A matroid approach to finding edge connectivity and packing arborescences, J. Comput. System Sci., 50 (1995), pp. 259--273.
25.
H. N. Gabow and R. E. Tarjan, Faster scaling algorithms for network problems, SIAM J. Comput., 18 (1989), pp. 1013--1036.
26.
P. Gawrychowski, S. Mozes, and O. Weimann, Submatrix maximum queries in Monge matrices are equivalent to predecessor search, in Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming, B. Speckmann, ed., Springer, Berlin, 2015, pp. 580--592.
27.
A. V. Goldberg, Scaling algorithms for the shortest paths problem, SIAM J. Comput., 24 (1995), pp. 494--504.
28.
R. E. Gomory and T. C. Hu, Multi-terminal network flows, J. SIAM, 9 (1961), pp. 551--570.
29.
M. T. Goodrich, Planar separators and parallel polygon triangulation, J. Comput. System Sci., 51 (1995), pp. 374--389.
30.
J. Hao and J. B. Orlin, A faster algorithm for finding the minimum cut in a directed graph, J. Algorithms, 17 (1994), pp. 424--446.
31.
M. R. Henzinger, P. N. Klein, S. Rao, and S. Subramanian, Faster shortest-path algorithms for planar graphs, J. Comput. System Sci., 55 (1997), pp. 3--23.
32.
A. Itai and M. Rodeh, Finding a minimum circuit in a graph, SIAM J. Comput., 7 (1978), pp. 413--423.
33.
G. F. Italiano, Y. Nussbaum, P. Sankowski, and C. Wulff-Nilsen, Improved algorithms for min cut and max flow in undirected planar graphs, in Proceedings of the 43rd ACM Symposium on Theory of Computing, ACM, New York, 2011, pp. 313--322.
34.
L. Janiga and V. Koubek, Minimum cut in directed planar networks, Kybernetika (Prague), 28 (1992), pp. 37--49.
35.
H. Kaplan, S. Mozes, Y. Nussbaum, and M. Sharir, Submatrix maximum queries in Monge matrices and Monge partial matrices, and their applications, in Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2012, pp. 338--355.
36.
H. Kaplan and Y. Nussbaum, Minimum $s$-$t$ cut in undirected planar graphs when the source and the sink are close, in Proceedings of the 28th International Symposium on Theoretical Aspects of Computer Science, T. Schwentick and C. Dürr, eds., Schloss Dagstuhl--Leibniz-Zentrum für Informatik, Wadern, Germany, 2011, pp. 117--128.
37.
D. R. Karger, Minimum cuts in near-linear time, J. ACM, 47 (2000), pp. 46--76.
38.
K.-I. Kawarabayashi and M. Thorup, Deterministic global minimum cut of a simple graph in near-linear time, in Proceedings of the 47th ACM Symposium on Theory of Computing, ACM, New York, 2015, pp. 665--674.
39.
S. Khuller and J. Naor, Flow in planar graphs: A survey of recent results, in Planar Graphs, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 9, AMS, New York, 1993, pp. 59--84.
40.
P. N. Klein, Multiple-source shortest paths in planar graphs, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2005, pp. 146--155.
41.
P. N. Klein, S. Mozes, and C. Sommer, Structured recursive separator decompositions for planar graphs in linear time, in Proceedings of the 45th ACM Symposium on Theory of Computing, ACM, New York, 2013, pp. 505--514.
42.
P. N. Klein, S. Mozes, and O. Weimann, Shortest paths in directed planar graphs with negative lengths: A linear-space $O(n\log^2 n)$-time algorithm, ACM Trans. Algorithms, 6 (2010), pp. 30.
43.
J. Łacki, Y. Nussbaum, P. Sankowski, and C. Wulff-Nilsen, Single source - all sinks max flows in planar digraphs, in Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2012, pp. 599--608.
44.
J. Łacki and P. Sankowski, Min-cuts and shortest cycles in planar graphs in $O(n\log\log n)$ time, in Proceedings of the 19th Annual European Symposium on Algorithms, Lecture Notes in Comput. Sci. 6942, Springer, Berlin, 2011, pp. 155--166.
45.
H.-C. Liang, Minimum Cuts and Shortest Cycles in Directed Planar Graphs via Shortest Non-crossing Paths, Master's thesis, National Taiwan University, Taipei, Taiwan, 2015.
46.
A. Lingas and E.-M. Lundell, Efficient approximation algorithms for shortest cycles in undirected graphs, Inform. Process. Lett., 109 (2009), pp. 493--498.
47.
R. J. Lipton and R. E. Tarjan, A separator theorem for planar graphs, SIAM J. Appl. Math., 36 (1979), pp. 177--189.
48.
B. Monien, The complexity of determining a shortest cycle of even length, Computing, 31 (1983), pp. 355--369.
49.
R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press, Cambridge, 1995.
50.
S. Mozes, C. Nikolaev, Y. Nussbaum, and O. Weimann, Minimum Cut of Directed Planar Graphs in $O(n\log\log n)$ Time, preprint, arXiv:1512.02068, 2015.
51.
S. Mozes and C. Wulff-Nilsen, Shortest paths in planar graphs with real lengths in $O(n\log^2n/\log\log n)$ time, in Proceedings of the 18th Annual European Symposium on Algorithms, M. de Berg and U. Meyer, eds., Lecture Notes in Comput. Sci. 6347, Springer, Berlin, 2010, pp. 206--217.
52.
K. Mulmuley, U. V. Vazirani, and V. V. Vazirani, Matching is as easy as matrix inversion, Combinatorica, 7 (1987), pp. 105--113.
53.
H. Nagamochi and T. Ibaraki, Computing edge-connectivity in multigraphs and capacitated graphs, SIAM J. Discrete Math., 5 (1992), pp. 54--66.
54.
J. B. Orlin, Max flows in $O(nm)$ time, or better, in Proceedings of the 45th ACM Symposium on Theory of Computing, ACM, New York, 2013, pp. 765--774.
55.
E. Papadopoulou, $k$-pairs non-crossing shortest paths in a simple polygon, Internat. J. Comput. Geom. Appl., 9 (1999), pp. 533--552.
56.
V. Polishchuk and J. S. B. Mitchell, Thick non-crossing paths and minimum-cost flows in polygonal domains, in Proceedings of the 23rd ACM Symposium on Computational Geometry, ACM, New York, 2007, pp. 56--65.
57.
J. H. Reif, Minimum $s$-$t$ cut of a planar undirected network in $O(n\log^2{n})$ time, SIAM J. Comput., 12 (1983), pp. 71--81.
58.
L. Roditty and R. Tov, Approximating the girth, ACM Trans. Algorithms, 9 (2013), pp. 15.
59.
L. Roditty and V. Vassilevska Williams, Subquadratic time approximation algorithms for the girth, in Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2012, pp. 833--845.
60.
M. Stoer and F. Wagner, A simple min-cut algorithm, J. ACM, 44 (1997), pp. 585--591.
61.
J. Takahashi, H. Suzuki, and T. Nishizeki, Finding shortest non-crossing rectilinear paths in plane regions, in Proceedings of the 4th International Symposium on Algorithms and Computation, Springer, Berlin, 1993, pp. 98--107.
62.
J.-Y. Takahashi, H. Suzuki, and T. Nishizeki, Shortest noncrossing paths in plane graphs, Algorithmica, 16 (1996), pp. 339--357.
63.
V. Vassilevska Williams, Multiplying matrices faster than Coppersmith-Winograd, in Proceedings of the 44th ACM Symposium on Theory of Computing, ACM, New York, 2012, pp. 887--898.
64.
V. Vassilevska Williams and R. Williams, Subcubic equivalences between path, matrix and triangle problems, in Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2010, pp. 645--654.
65.
K. Weihe, Edge-disjoint $(s, t)$-paths in undirected planar graphs in linear time, J. Algorithms, 23 (1997), pp. 121--138.
66.
O. Weimann and R. Yuster, Computing the girth of a planar graph in $O(n\log n)$ time, SIAM J. Discrete Math., 24 (2010), pp. 609--616.
67.
C. Wulff-Nilsen, Algorithms for Planar Graphs and Graphs in Metric Spaces, PhD thesis, University of Copenhagen, Copenhagen, 2010.
68.
R. Yuster, A shortest cycle for each vertex of a graph, Inform. Process. Lett., 111 (2011), pp. 1057--1061.
69.
R. Yuster and U. Zwick, Finding even cycles even faster, SIAM J. Discrete Math., 10 (1997), pp. 209--222.
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© 2017, Society for Industrial and Applied Mathematics.
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Submitted: 22 January 2016
Accepted: 18 November 2016
Published online: 7 March 2017
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Ministry of Science and Technology, Taiwan http://dx.doi.org/10.13039/501100004663 : 104-2221-E-002-044-MY3
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