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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References
Ball, M.O., Provan, J.S.: Calculating bounds on reachability and connectnedness in stochastic networks. Networks 13, 253–278 (1983)
Bezáková, I., Friedlander, A.J.: Counting and sampling minimum (s,t)-cuts in weighted planar graphs in polynomial time. Theor. Comp. Sci. 417, 2–11 (2012)
Borradaile, G., Klein, P.N.: An O(nlogn) algorithm for maximum st-flow in a directed planar graph. J. ACM 56(2) (2009)
Borradaile, G., Klein, P.N., Mozes, S., Nussbaum, Y., Wulff-Nilsen, C.: Multiple-source multiple-sink maximum flow in directed planar graphs in near-linear time. In: Proceedings of the 52nd IEEE Symposium on Foundations of Computer Science (FOCS), pp. 170–179 (2011)
Borradaile, G., Sankowski, P., Wulff-Nilsen, C.: Min st-cut oracle for planar graphs with near-linear preprocessing time. In: Proceedings of the 51st IEEE Symposium on Foundations of Computer Science (FOCS), pp. 601–610 (2010)
Boykov, Y., Veksler, O.: Graph cuts in vision and graphics: Theories and applications. In: Paragios, N., Chen, Y., Faugeras, O. (eds.) Handbook of Mathematical Models in Computer Vision. Springer (2006)
Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Trans. Pattern Anal. Mach. Intell. 23(11), 1222–1239 (2001)
Creed, P.: Counting and sampling problems on Eulerian graphs. Ph.D. Dissertation, University of Edinburgh (2010)
Ge, Q., Štefankovič, D.: The complexity of counting Eulerian tours in 4-regular graphs. Algorithmica 63(3), 588–601 (2012)
Grady, L.: Minimal surfaces extend shortest path segmentation methods to 3D. IEEE Trans. Pattern Anal. Mach. Intell. 32(2), 321–334 (2010)
Italiano, G.F., Nussbaum, Y., Sankowski, P., Wulff-Nilsen, C.: Improved algorithms for min cut and max flow in undirected planar graphs. In: Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC), pp. 313–322 (2011)
Liskiewicz, M., Ogihara, M., Toda, S.: The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes. Theoretical Computer Science 304(1-3), 129–156 (2003)
Provan, J.S., Ball, M.O.: The complexity of counting cuts and of computing the probability that a graph is connected. SIAM J. Comput. 12(4), 777–788 (1983)
Vicente, S., Kolmogorov, V., Rother, C.: Graph cut based image segmentation with connectivity priors. In: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR (2008)
Zeng, Y., Samaras, D., Chen, W., Peng, Q.: Topology cuts: A novel min-cut/max-flow algorithm for topology preserving segmentation in n-d images. Computer Vision Image Understanding 112, 81–90 (2008)
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Bezáková, I., Langley, Z. (2012). Contiguous Minimum Single-Source-Multi-Sink Cuts in Weighted Planar Graphs. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds) Computing and Combinatorics. COCOON 2012. Lecture Notes in Computer Science, vol 7434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32241-9_5
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