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

Mozes, Shay; Nikolaev, K.; Nussbaum, Yahav; Weimann, Oren

doi:10.1137/1.9781611975031.32

Cited by 9 publications

(15 citation statements)

References 25 publications

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“…As discussed in the introduction, our perturbation scheme can be used in a black box fashion to immediately derandomize the O(n log log n) time minimum cut algorithm of Mozes et al [53] for directed planar graphs. The only change necessary to derandomize their algorithm is to guarantee uniqueness of shortest paths in the dual graph.…”

Section: Minimum Cut In Directed Planar Graphsmentioning

confidence: 99%

“…The most commonly applied consequence of this assumption is that the shortest paths entering (or leaving) a common vertex do not cross one another. From this consequence, one can prove near-linear running time bounds for a variety of problems, including the computation of maximum flows [4,5,24,26] and global minimum cuts [53] in directed planar (genus 0) graphs as well as the computation of minimum cut oracles in planar and more general embedded graphs [3,6] (see also ).This assumption is also used in algorithms for the multiple-source shortest paths problem introduced for planar graphs by Klein [46]. In the multiple-source shortest paths problem, one is given a surface embedded graph G = (V, E, F ) of genus g with vertices V , edges E, and faces F .…”

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confidence: 99%

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Holiest minimum-cost paths and flows in surface graphs

Erickson

Fox²,

Lkhamsuren³

2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

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Let G be an edge-weighted directed graph with n vertices embedded on an orientable surface of genus g. We describe a simple deterministic lexicographic perturbation scheme that guarantees uniqueness of minimum-cost flows and shortest paths in G. The perturbations take O(gn) time to compute. We use our perturbation scheme in a black box manner to derive a deterministic O(n log log n) time algorithm for minimum cut in directed edge-weighted planar graphs and a deterministic O(g 2 n log n) time proprocessing scheme for the multiple-source shortest paths problem of computing a shortest path oracle for all vertices lying on a common face of a surface embedded graph. The latter result yields faster deterministic near-linear time algorithms for a variety of problems in constant genus surface embedded graphs.Finally, we open the black box in order to generalize a recent linear-time algorithm for multiplesource shortest paths in unweighted undirected planar graphs to work in arbitrary orientable surfaces. Our algorithm runs in O(g 2 n log g) time in this setting, and it can be used to give improved linear time algorithms for several problems in unweighted undirected surface embedded graphs of constant genus including the computation of minimum cuts, shortest topologically non-trivial cycles, and minimum homology bases.Many recent combinatorial optimization algorithms for directed surface embedded graphs rely on a common assumption: the shortest path between any pair of vertices is unique. The most commonly applied consequence of this assumption is that the shortest paths entering (or leaving) a common vertex do not cross one another. From this consequence, one can prove near-linear running time bounds for a variety of problems, including the computation of maximum flows [4,5,24,26] and global minimum cuts [53] in directed planar (genus 0) graphs as well as the computation of minimum cut oracles in planar and more general embedded graphs [3,6] (see also ).This assumption is also used in algorithms for the multiple-source shortest paths problem introduced for planar graphs by Klein [46]. In the multiple-source shortest paths problem, one is given a surface embedded graph G = (V, E, F ) of genus g with vertices V , edges E, and faces F . The goal is to compute a representation of all shortest paths from vertices on a common face r ∈ F to all other vertices in the graph. Assuming uniqueness of shortest paths, multiple-source shortest paths can be computed in only O(g n log n) time [11,46]. Algorithms for this problem can be used to solve a variety of problems in planar and more general surface embedded graphs of constant genus in near-linear time. Such results include the computation of shortest cycles with non-trivial topology [2,11,27,30,32,34], the computation of maximum flows and minimum cuts [5,14,16,28,30,41,49], the computation of exact and approximate distance oracles [10,44,54], and even the computation of single-source shortest paths [47,55].Enforcing uniqueness. Unfortunately, it is often difficult to actually enfo...

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Section: Minimum Cut In Directed Planar Graphsmentioning

confidence: 99%

mentioning

confidence: 99%

Holiest minimum-cost paths and flows in surface graphs

Erickson

Fox²,

Lkhamsuren³

2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

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“…The best known algorithms for both problems, due to Italiano et al [33], run in O(n log log n) time. The attempt of Janiga and Koubek [34] to generalize Reif's algorithm to directed planar G turned out to be flawed [17,36,51]. Borradaile and Klein [1] and Erickson [14] gave O(n log n)time algorithms for both problems on directed planar graphs.…”

Section: Related Workmentioning

confidence: 99%

“…As a matter of fact, this shortest-noncrossing-paths problem can be solved by the O(n log n)-time algorithm of Klein [40], already yielding improved O(n log 2 n)-time algorithms for the minimum-cut and shortest-cycle problems. (Mozes et al [51] also mentioned that O(n log 2 n)-time algorithms can be obtained by plugging in the O(n log n)-time minimum st-cut algorithm of Borradaile and Klein [1] into a directed version of the reduction algorithm of Chalermsook et al [8].) To achieve the time complexity of Theorem 1.1, Section 5 solves the problem in O(n log log n) time by extending the algorithm of Italiano et al [33] for an undi-rected plane graph.…”

Section: Technical Overview and Outlinementioning

confidence: 99%

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Minimum Cuts and Shortest Cycles in Directed Planar Graphs via Noncrossing Shortest Paths

Liang¹,

Lu²

2017

SIAM J. Discrete Math.

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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 Łącki, 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 non-degenerate 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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Topologically Trivial Closed Walks in Directed Surface Graphs

Erickson

Wang

2020

Discrete Comput Geom

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Let G be a directed graph with n vertices and m edges, embedded on a surface S, possibly with boundary, with first Betti number β. We consider the complexity of finding closed directed walks in G that are either contractible (trivial in homotopy) or bounding (trivial in integer homology) in S. Specifically, we describe algorithms to determine whether G contains a simple contractible cycle in O(n + m) time, or a contractible closed walk in O(n + m) time, or a bounding closed walk in O(β(n + m)) time. Our algorithms rely on subtle relationships between strong connectivity in G and in the dual graph G ; our contractible-closed-walk algorithm also relies on a seminal topological result of Hass and Scott. We also prove that detecting simple bounding cycles is NP-hard.We also describe three polynomial-time algorithms to compute shortest contractible closed walks, depending on whether the fundamental group of the surface is free, abelian, or hyperbolic. A key step in our algorithm for hyperbolic surfaces is the construction of a context-free grammar with O(g 2 L 2 ) non-terminals that generates all contractible closed walks of length at most L, and only contractible closed walks, in a system of quads of genus g ≥ 2. Finally, we show that computing shortest simple contractible cycles, shortest simple bounding cycles, and shortest bounding closed walks are all NP-hard.

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

Cited by 9 publications

References 25 publications

Holiest minimum-cost paths and flows in surface graphs

Holiest minimum-cost paths and flows in surface graphs

Minimum Cuts and Shortest Cycles in Directed Planar Graphs via Noncrossing Shortest Paths

Topologically Trivial Closed Walks in Directed Surface Graphs

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