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Encyclopedia of Algorithms pp 1576–1579Cite as

Planar Maximum s- t Flow

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  • First Online:

  • 65 Accesses

Years and Authors of Summarized Original Work

  • 2006; Borradaile, Klein

  • 2009; Borradaile, Klein

  • 2010; Erickson

Problem Definition

Given a directed, planar graph G = (V, E) with arc capacities \(c : E \rightarrow \mathfrak{R}^{+}\), a source vertex s, and a sink vertex t, the goal is to find a flow assignment f e for each arc e ∈ E such that

$$\displaystyle\begin{array}{rcl} \max & & \sum _{su:su\in E}f_{su} \\ \mathrm{s.t.}& & \sum _{uv:uv\in E}f_{uv} -\sum _{vw:vw\in E}f_{vw} = 0 \\ & & \qquad \forall v \in V \setminus \{s,t\} {}\end{array}$$
(1)
$$\displaystyle\begin{array}{rcl} & & 0 \leq f_{e} \leq c_{e}\qquad \forall e \in E{}\end{array}$$
(2)

Key Results

In the paper proposing the maximum flow problem in general graphs, Ford and Fulkerson [5] gave a generic method for computing a maximum flow: the augmenting-path algorithm. The algorithm is iterative: find a path P from the source to the sink such that capacity constraint (2) is loose for each arc on P (residual); increase the...

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Recommended Reading

  1. Coupry L (1997) A simple linear algorithm for the edge-disjoint (s,t)-paths problem in undirected planar graphs. Inf Process Lett 64:83–86

    Article  MathSciNet  Google Scholar 

  2. Eisenstat D (2013) Trickle: linear-time maximum flow in planar graphs with unit capacities (java). http://www.davideisenstat.com/trickle/

  3. Erickson J (2010) Maximum flows and parametric shortest paths in planar graphs. In: 21st SODA, Austin, pp 794–804

    Google Scholar 

  4. Fakcharoenphol J, Rao S (2006) Planar graphs, negative weight edges, shortest paths, and near linear time. J Comput Syst Sci 72(5):868–889. doi:http://dx.doi.org/10.1016/j.jcss.2005.05.007

  5. Ford C, Fulkerson D (1956) Maximal flow through a network. Can J Math 8:399–404

    Article  MathSciNet  MATH  Google Scholar 

  6. Greig D, Porteous B, Seheult A (1989) Exact maximum a posteriori estimation for binary images. J R Stat Soc B 51(2):271–279

    Google Scholar 

  7. Hassin R (1981) Maximum flow in (s, t) planar networks. Inf Process Lett 13(3):107

    Article  MathSciNet  Google Scholar 

  8. Hassin R, Johnson DB (1985) An \(O(n\log ^{2}n)\) algorithm for maximum flow in undirected planar networks. SIAM J Comput 14:612–624. doi:http://locus.siam.org/SICOMP/volume-14/art_0214045.html

  9. Henzinger MR, Klein PN, Rao S, Subramanian S (1997) Faster shortest-path algorithms for planar graphs. J Comput Syst Sci 55(1):3–23. doi:10.1145/195058.195092

    MathSciNet  MATH  Google Scholar 

  10. Hoch J, Wang J (2012) Max flow in a directed planar graph. https://github.com/jrshoch/msmsmaxflow

  11. Itai A, Shiloach Y (1979) Maximum flow in planar networks. SIAM J Comput 8:135–150

    Article  MathSciNet  MATH  Google Scholar 

  12. Italiano GF, Nussbaum Y, Sankowski P, Wulff-Nilsen C (2011) Improved algorithms for min cut and max flow in undirected planar graphs. In: 43rd STOC, San Jose, pp 313–322

    Google Scholar 

  13. Johnson DB, Venkatesan SM (1983) Partition of planar flow networks. In: 24th FOCS, Tucson, pp 259–264. doi:10.1109/SFCS.1983.44

    Google Scholar 

  14. Khuller S, Naor J, Klein P (1993) The lattice structure of flow in planar graphs. SIAM J Discret Math 6(3):477–490. doi:10.1137/0406038

    MathSciNet  MATH  Google Scholar 

  15. Klein PN (2005) Multiple-source shortest paths in planar graphs. In: 16th SODA, Vancouver, pp 146–155. doi:10.1145/1070454

    Google Scholar 

  16. Klein PN, Mozes S, Weimann O (2010) Shortest paths in directed planar graphs with negative lengths: a linear-space \(O(n\log ^{2}n)\)-time algorithm. TALG 6(2):1–18. doi:http://doi.acm.org/10.1145/1721837.1721846

  17. Miller GL, Naor J (1995) Flow in planar graphs with multiple sources and sinks. SIAM J Comput 24(5):1002–1017. doi:10.1137/S0097539789162997

    Article  MathSciNet  MATH  Google Scholar 

  18. Mozes S, Wulff-Nilsen C (2010) Shortest paths in planar graphs with real lengths in \(O(n\log ^{2}n/\log \log n)\) time. In: 18th ESA, Liverpool, pp 206–217

    Google Scholar 

  19. Reif J (1983) Minimum s-t cut of a planar undirected network in \(O(n\log ^{2}n)\) time. SIAM J Comput 12:71–81. doi:http://locus.siam.org/SICOMP/volume-12/art_0212005.html

  20. Schmidt FR, Toeppe E, Cremers D (2009) Efficient planar graph cuts with applications in computer vision. In: CVPR, Miami, pp 351–356

    Google Scholar 

  21. Weihe K (1994) Edge-disjoint (s,t)-paths in undirected planar graphs in linear time. In: Proceedings of the European symposium on algorithms, Edinburgh. LNCS, vol 855, pp 130–137

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, Brown University, Providence, RI, USA

    Glencora Borradaile

  2. School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR, USA

    Glencora Borradaile

Authors
  1. Glencora Borradaile
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Correspondence to Glencora Borradaile .

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Borradaile, G. (2016). Planar Maximum s- t Flow. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_626

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