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Min st-Cut Oracle for Planar Graphs with Near-Linear Preprocessing Time

Published: 13 January 2015 Publication History

Abstract

For an undirected n-vertex planar graph G with nonnegative edge weights, we consider the following type of query: given two vertices s and t in G, what is the weight of a min st-cut in G? We show how to answer such queries in constant time with O(n log4 n) preprocessing time and O(n log n) space. We use a Gomory-Hu tree to represent all the pairwise min cuts implicitly. Previously, no subquadratic time algorithm was known for this problem. Since all-pairs min cut and the minimum-cycle basis are dual problems in planar graphs, we also obtain an implicit representation of a minimum-cycle basis in O(n log4 n) time and O(n log n) space. Additionally, an explicit representation can be obtained in O(C) time and space where C is the size of the basis.
These results require that shortest paths are unique. This can be guaranteed either by using randomization without overhead or deterministically with an additional log2 n factor in the preprocessing times.

References

[1]
S. Alstrup, J. Holm, K. de Lichtenberg, and M. Thorup. 2005. Maintaining information in fully dynamic trees with top trees. ACM Transactions on Algorithms 1, 2 (2005), 243--264.
[2]
E. Amaldi, C. Iuliano, T. Jurkiewicz, K. Mehlhorn, and R. Rizzi. 2009. Breaking the O(m2n) barrier for minimum cycle bases. In Proceedings of the 17th European Symposium on Algorithms (Lecture Notes in Computer Science), A. Fiat and P. Sanders (Eds.). 301--312.
[3]
A. Bhalgat, R. Hariharan, D. Panigrahi, and K. Telikepalli. 2007. An Õ(mn) Gomory-Hu tree construction algorithm for unweighted graphs. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing. 605--614.
[4]
J. Fakcharoenphol and S. Rao. 2006. Planar graphs, negative weight edges, shortest paths, and near linear time. Journal of Computer and System Sciences 72, 5 (2006), 868--889.
[5]
R. Gomory and T. Hu. 1961. Multi-terminal network flows. Journal of SIAM 9, 4 (1961), 551--570.
[6]
D. Hartvigsen and R. Mardon. 1994. The all-pairs min cut problem and the minimum cycle basis problem on planar graphs. SIAM Journal on Discrete Mathematics 7, 3 (1994), 403--418.
[7]
M. R. Henzinger, P. N. Klein, S. Rao, and S. Subramanian. 1997. Faster shortest-path algorithms for planar graphs. Journal of Computer and System Sciences 55, 1 (1997), 3--23.
[8]
J. Horton. 1987. A polynomial time algorithm to find the shortest cycle basis of a graph. SIAM Journal on Computing 16 (1987), 358--366.
[9]
A. Itai and Y. Shiloach. 1979. Maximum flow in planar networks. SIAM Journal on Computing 8 (1979), 135--150.
[10]
G. Italiano, Y. Nussbaum, P. Sankowski, and C. Wulff-Nilsen. 2011. Improved algorithms for min cut and max flow in undirected planar graphs. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (STOC'11). ACM, New York, NY, 313--322.
[11]
G. Kant and H. Bodlaender. 1992. Triangulating planar graphs while minimizing the maximum degree. In Algorithm Theory -- SWAT'92, Otto Nurmi and Esko Ukkonen (Eds.). Lecture Notes in Computer Science, Vol. 621. Springer Berlin/Heidelberg, 258--271.
[12]
H. Kaplan and N. Shafrir. 2008. Path minima in incremental unrooted trees. In Proceedings of the 16th European Symposium on Algorithms (Lecture Notes in Computer Science). 565--576.
[13]
G. Kirchhoff. 1847. Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Poggendorf Ann. Physik 72 (1847), 497--508. English transl. in Trans. Inst. Radio Engrs. CT-5 (1958), pp. 4--7.
[14]
P. Klein, S. Mozes, and C. Sommer. 2013. Structured recursive separator decompositions for planar graphs in linear time. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing.
[15]
P. N. Klein. 2005. Multiple-source shortest paths in planar graphs. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms. 146--155.
[16]
D. E. Knuth. 1968. The Art of Computer Programming. Vol. 1. Addison-Wesley.
[17]
G. L. Miller. 1986. Finding small simple cycle separators for 2-connected planar graphs. Journal of Computer and System Sciences 32, 3 (1986), 265--279.
[18]
K. Mulmuley, V. Vazirani, and U. Vazirani. 1987. Matching is as easy as matrix inversion. Combinatorica 7, 1 (1987), 345--354.
[19]
J. Reif. 1983. Minimum s-t cut of a planar undirected network in O(n log2 n) time. SIAM Journal on Computing 12, 1 (1983), 71--81.
[20]
H. Whitney. 1933. Planar graphs. Fundamenta Mathematicae 21 (1933), 73--84.

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    Published In

    cover image ACM Transactions on Algorithms
    ACM Transactions on Algorithms  Volume 11, Issue 3
    January 2015
    269 pages
    ISSN:1549-6325
    EISSN:1549-6333
    DOI:10.1145/2721890
    Issue’s Table of Contents
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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    Publication History

    Published: 13 January 2015
    Accepted: 01 September 2013
    Revised: 01 June 2013
    Received: 01 November 2012
    Published in TALG Volume 11, Issue 3

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    Author Tags

    1. Minimum cut
    2. minimum cycle basis
    3. planar graphs

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    Cited By

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    • (2023)Almost Optimal Exact Distance Oracles for Planar GraphsJournal of the ACM10.1145/358047470:2(1-50)Online publication date: 25-Mar-2023
    • (2023)All-Pairs Max-Flow is no Harder than Single-Pair Max-Flow: Gomory-Hu Trees in Almost-Linear Time2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00137(2204-2212)Online publication date: 6-Nov-2023
    • (2023)Some Insights on Dynamic Maintenance of Gomory-Hu Tree in Cactus Graphs and General GraphsAlgorithms and Discrete Applied Mathematics10.1007/978-3-031-25211-2_18(231-244)Online publication date: 9-Feb-2023
    • (2022)Exact Distance Oracles for Planar Graphs with Failing VerticesACM Transactions on Algorithms10.1145/351154118:2(1-23)Online publication date: 30-Mar-2022
    • (2022)Breaking the Cubic Barrier for All-Pairs Max-Flow: Gomory-Hu Tree in Nearly Quadratic Time2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00088(884-895)Online publication date: Oct-2022
    • (2022)APMF < APSP? Gomory-Hu Tree for Unweighted Graphs in Almost-Quadratic Time2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00112(1135-1146)Online publication date: Feb-2022
    • (2022)Single-source shortest paths and strong connectivity in dynamic planar graphsJournal of Computer and System Sciences10.1016/j.jcss.2021.09.008124:C(97-111)Online publication date: 1-Mar-2022
    • (2021)A deterministic parallel APSP algorithm and its applicationsProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458081(255-272)Online publication date: 10-Jan-2021
    • (2021)Subcubic algorithms for Gomory–Hu tree in unweighted graphsProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451073(1725-1737)Online publication date: 15-Jun-2021
    • (2020)Cut-Equivalent Trees are Optimal for Min-Cut Queries2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS46700.2020.00019(105-118)Online publication date: Nov-2020
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