Glencora Borradaile
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.
- 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. DOI:http://dx.doi.org/10.1145/1103963.1103966 Google Scholar
Digital Library
- 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.Google Scholar
- 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. DOI:http://dx.doi.org/10.1145/1250790.1250879 Google ScholarDigital Library
- 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. DOI:http://dx.doi.org/10.1016/j.jcss.2005.05.007 Google ScholarDigital Library
- R. Gomory and T. Hu. 1961. Multi-terminal network flows. Journal of SIAM 9, 4 (1961), 551--570. DOI:http://dx.doi.org/10.1137/0109047Google Scholar
- 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. DOI:http://dx.doi.org/10.1137/S0895480190177042 Google ScholarDigital Library
- 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. DOI:http://dx.doi.org/10.1145/195058.195092 Google ScholarDigital Library
- J. Horton. 1987. A polynomial time algorithm to find the shortest cycle basis of a graph. SIAM Journal on Computing 16 (1987), 358--366. Google ScholarDigital Library
- A. Itai and Y. Shiloach. 1979. Maximum flow in planar networks. SIAM Journal on Computing 8 (1979), 135--150.Google ScholarCross Ref
- 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. DOI:http://dx.doi.org/10.1145/1993636.1993679 Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- 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. Google ScholarDigital Library
- 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. DOI:http://dx.doi.org/10.1145/1070454 Google ScholarDigital Library
- D. E. Knuth. 1968. The Art of Computer Programming. Vol. 1. Addison-Wesley.Google Scholar
- 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. DOI:http://dx.doi.org/10.1016/0022-0000(86)90030-9 Google ScholarDigital Library
- K. Mulmuley, V. Vazirani, and U. Vazirani. 1987. Matching is as easy as matrix inversion. Combinatorica 7, 1 (1987), 345--354. Google ScholarDigital Library
- 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. DOI:http://dx.doi.org/SICOMP/volume-12/art_0212005.htmlGoogle ScholarCross Ref
- H. Whitney. 1933. Planar graphs. Fundamenta Mathematicae 21 (1933), 73--84.Google ScholarCross Ref
Index Terms
- Min st-Cut Oracle for Planar Graphs with Near-Linear Preprocessing Time
Recommendations
Min st-cut Oracle for Planar Graphs with Near-Linear Preprocessing Time
FOCS '10: Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer ScienceFor an undirected $n$-vertex planar graph $G$ with non-negative 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 ...
Deterministic Global Minimum Cut of a Simple Graph in Near-Linear Time
STOC '15: Proceedings of the forty-seventh annual ACM symposium on Theory of ComputingWe present a deterministic near-linear time algorithm that computes the edge-connectivity and finds a minimum cut for a simple undirected unweighted graph G with n vertices and m edges. This is the first o(mn) time deterministic algorithm for the ...
Optimal EREW parallel algorithms for connectivity, ear decomposition and st-numbering of planar graphs
IPPS '91: Proceedings of the Fifth International Parallel Processing SymposiumParallel EREW deterministic algorithms for finding the connected components, ear decomposition and st-numbering of a planar graph are presented. The algorithms take O(log(n)) time with /sub log(n)///sup n+m/ processors. Previous results have the same ...
Comments