Improved algorithms for min cut and max flow in undirected planar graphs

Italiano, Giuseppe F.; Nussbaum, Yahav; Sankowski, Piotr; Wulff-Nilsen, Christian

doi:10.1145/1993636.1993679

Cited by 92 publications

(160 citation statements)

References 22 publications

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“…Up to now, the fastest known algorithm computing an r-division with a constant number of holes per region runs in time O(n log r + (n/ √ r) log n) [INSWN11]. This makes it one of the time bottlenecks in the state-of-the-art algorithms for minimum st-cut and maximum st-flow [INSWN11] and minimum cut [LS11] in undirected planar graphs and bounded-genus graphs [EFN12].…”

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confidence: 99%

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Structured recursive separator decompositions for planar graphs in linear time

Klein

Mozes

Sommer

2013

Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing

112

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Given a planar graph G on n vertices and an integer parameter r < n, an r-division of G with few holes is a decomposition of G into O(n/r) regions of size at most r such that each region contains at most a constant number of faces that are not faces of G (also called holes), and such that, for each region, the total number of vertices on these faces is O( √ r). We provide a linear-time algorithm for computing r-divisions with few holes. In fact, our algorithm computes a structure, called decomposition tree, which represents a recursive decomposition of G that includes r-divisions for essentially all values of r. In particular, given an increasing sequence r = (r1, r2, ...), our algorithm can produce a recursive r-division with few holes in linear time.r-divisions with few holes have been used in efficient algorithms to compute shortest paths, minimum cuts, and maximum flows. Our linear-time algorithm improves upon the decomposition algorithm used in the state-of-the-art algorithm for minimum st-cut (Italiano, Nussbaum, Sankowski, and Wulff-Nilsen, STOC 2011), removing one of the bottlenecks in the overall running time of their algorithm (analogously for minimum cut in planar and bounded-genus graphs).

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confidence: 99%

“…This makes it one of the time bottlenecks in the state-of-the-art algorithms for minimum st-cut and maximum st-flow [INSWN11] and minimum cut [LS11] in undirected planar graphs and bounded-genus graphs [EFN12]. Whether such an r-division can be computed in linear time was an open problem until the current work.…”

mentioning

confidence: 99%

Structured recursive separator decompositions for planar graphs in linear time

Klein

Mozes

Sommer

2013

Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing

112

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“…Of course, much work is also dedicated to computing cuts and flows in planar graphs, both undirected [11,28,31,35,36,43,55] and directed [6,58]. The current best algorithms run in O(n log log n) time in undirected graphs [36] and O(n log n) time in directed graphs [6], considerably improving on running times for arbitrary sparse graphs.…”

Section: Flows and Cuts In Restrictive Graph Familiesmentioning

confidence: 99%

Counting and sampling minimum cuts in genus g graphs

Chambers

Fox

Nayyeri

2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

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Let G be a directed graph with n vertices embedded on an orientable surface of genus g with two designated vertices s and t. We show that computing the number of minimum (s, t)-cuts in G is fixed-parameter tractable in g. Specifically, we give a 2 O(g) n 2 time algorithm for this problem. Our algorithm requires counting sets of cycles in a particular integer homology class. That we can count these cycles is an interesting result in itself as there are no prior results that are fixed-parameter tractable and deal directly with integer homology. We also describe an algorithm which, after running our algorithm to count minimum cuts once, can sample an (s, t)-minimum cut uniformly at random in O(n log n) time per sample.

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“…Borradaile, Sankowski, and Wulff-Nilsen [5] produce a minimum single-source-single-sink cut for any source-sink pair in time proportional to the size of the cut, after an initial O(n polylog n) preprocessing time. Italiano, Nussbaum, Sankowski, and Wulff-Nilsen [11] give algorithms for undirected planar graphs that break the O(n log n) time barrier. Recently, Borradaile, Klein, Mozes, Nussbaum, and Wulff-Nilsen [4] gave an O(n log 3 n) algorithm for maximum flow from multiple sources to multiple sinks.…”

Section: Introductionmentioning

confidence: 99%

Contiguous Minimum Single-Source-Multi-Sink Cuts in Weighted Planar Graphs

Bezáková

Langley

2012

Lecture Notes in Computer Science

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Abstract. We present a fast algorithm for uniform sampling of contiguous minimum cuts separating a source vertex from a set of sink vertices in a weighted undirected planar graph with n vertices embedded in the plane. The algorithm takes O(n) time per sample, after an initial O(n 3 ) preprocessing time during which the algorithm computes the number of all such contiguous minimum cuts. Contiguous cuts (that is, cuts where a naturally defined boundary around the cut set forms a simply connected planar region) have applications in computer vision and medical imaging [6,14].

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Improved algorithms for min cut and max flow in undirected planar graphs

Cited by 92 publications

References 22 publications

Structured recursive separator decompositions for planar graphs in linear time

Structured recursive separator decompositions for planar graphs in linear time

Counting and sampling minimum cuts in genus g graphs

Contiguous Minimum Single-Source-Multi-Sink Cuts in Weighted Planar Graphs

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