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11
Minimum Cuts and Shortest Homologous Cycles
 SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2009
"... We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the spec ..."
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Cited by 34 (12 self)
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We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the special case of planar graphs, for which O(n log n)time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimumcut algorithm computes a minimumcost subgraph in every Z2homology class. We also prove that finding a minimumcost subgraph homologous to a single input cycle is NPhard.
Minimum Cuts and Shortest NonSeparating Cycles via Homology Covers
 SYMPOSIUM ON DISCRETE ALGORITHMS
, 2011
"... Let G be a directed graph with weighted edges, embedded on a surface of genus g with b boundaries. We describe an algorithm to compute the shortest directed cycle in G in any given � 2homology class in 2 O(g+b) n log n time; this problem is NPhard even for undirected graphs. We also present two ap ..."
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Cited by 18 (5 self)
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Let G be a directed graph with weighted edges, embedded on a surface of genus g with b boundaries. We describe an algorithm to compute the shortest directed cycle in G in any given � 2homology class in 2 O(g+b) n log n time; this problem is NPhard even for undirected graphs. We also present two applications of our algorithm. The first is an algorithm to compute the shortest nonseparating directed cycle in G in 2 O(g) n log n time, improving the recent algorithm of Cabello et al. [SOCG 2010] for all g = o(log n). The second is a combinatorial algorithm to compute minimum (s, t)cuts in undirected surface graphs in 2 O(g) n log n time, improving an algorithm of Chambers et al. [SOCG 2009] for all positive g. Unlike earlier algorithms for surface graphs that construct and search finite portions of the universal cover, our algorithms use another canonical covering space, called the Z 2homology cover.
Finding shortest nontrivial cycles in directed graphs on surfaces
 In These Proceedings
, 2010
"... Let D be a weighted directed graph cellularly embedded in a surface of genus g, orientable or not, possibly with boundary. We describe algorithms to compute a shortest noncontractible and a shortest surface nonseparating cycle in D. This generalizes previous results that only dealt with undirected ..."
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Cited by 16 (3 self)
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Let D be a weighted directed graph cellularly embedded in a surface of genus g, orientable or not, possibly with boundary. We describe algorithms to compute a shortest noncontractible and a shortest surface nonseparating cycle in D. This generalizes previous results that only dealt with undirected graphs. Our first algorithm computes such cycles in O(n 2 log n) time, where n is the total number of vertices and edges of D, thus matching the complexity of the best known algorithm in the undirected case. It revisits and extends Thomassen’s 3path condition; the technique applies to other families of cycles as well. We also give an algorithm with subquadratic complexity in the complexity of the input graph, if g is fixed. Specifically, we can solve the problem in O ( √ g n 3/2 log n) time, using a divideandconquer technique that simplifies the graph while preserving the topological properties of its cycles. A variant runs in O(ng log g + nlog 2 n) for graphs of bounded treewidth.
Global Minimum Cuts in Surface Embedded Graphs
"... We give a deterministic algorithm to find the minimum cut in a surfaceembedded graph in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in g O(g) n log log n time, matching the running time of the fastest algorithm kno ..."
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Cited by 8 (5 self)
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We give a deterministic algorithm to find the minimum cut in a surfaceembedded graph in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in g O(g) n log log n time, matching the running time of the fastest algorithm known for planar graphs, due to Ł ˛acki and Sankowski, for any constant g. Indeed, our algorithm calls Ł ˛acki and Sankowski’s recent O(n log log n) time planar algorithm as a subroutine. Previously, the best time bounds known for this problem followed from two algorithms for general sparse graphs: a randomized algorithm of Karger that runs in O(n log³ n) time and succeeds with high probability, and a deterministic algorithm of Nagamochi and Ibaraki that runs in O(n² log n) time. We can also achieve a deterministic g O(g) n² log log n time bound by repeatedly applying the best known algorithm for minimum (s, t)cuts in surface graphs. The bulk of our work focuses on the case where the dual of the minimum cut splits the underlying surface into multiple components with positive genus.
Shortest nontrivial cycles in directed surface graphs
 In Proc. 27th Ann. Symp. Comput. Geom
, 2011
"... Let G be a directed graph embedded on a surface of genus g. We describe an algorithm to compute the shortest nonseparating cycle in G in O(g 2 n log n) time, exactly matching the fastest algorithm known for undirected graphs. We also describe an algorithm to compute the shortest noncontractible cy ..."
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Cited by 8 (2 self)
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Let G be a directed graph embedded on a surface of genus g. We describe an algorithm to compute the shortest nonseparating cycle in G in O(g 2 n log n) time, exactly matching the fastest algorithm known for undirected graphs. We also describe an algorithm to compute the shortest noncontractible cycle in G in g O(g) n log n time, matching the fastest algorithm for undirected graphs of constant genus.
Branching and Circular Features in High Dimensional Data
, 2011
"... Large observations and simulations in scientific research give rise to highdimensional data sets that present many challenges and opportunities in data analysis and visualization. Researchers in the application domains such as engineering, computational biology, climate study, imaging and motion ca ..."
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Cited by 6 (3 self)
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Large observations and simulations in scientific research give rise to highdimensional data sets that present many challenges and opportunities in data analysis and visualization. Researchers in the application domains such as engineering, computational biology, climate study, imaging and motion capture are faced with the problem of how to discover compact representations of highdimensional data while preserving their intrinsic structure. In many applications, the original data is projected onto lowdimensional space via dimensionality reduction techniques prior to modeling. One problem with this approach is that the projection step in the process can fail to preserve structure in the data that is only apparent in high dimensions. Conversely, such techniques may create structural illusions in the projection, implying structure not present in the original highdimensional data. Our solution is to utilize topological techniques to recover important structures in highdimensional data that contains nontrivial topology. Specifically, we are interested in two types of features in high dimensions: local branching structures and global circular structures. We construct local and global circlevalued coordinate functions to represent such features. Subsequently, we perform dimensionality reduction on the data while ensuring such structures are visually preserved. Our results reveal neverbeforeseen structures on realworld data sets from a variety of applications. Branching and Circular Features in High Dimensional Data
Shortest Nontrivial Cycles in Directed and Undirected Surface Graphs
"... Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of nontrivial cycles in G, using different techniques depending on whether or not G is an undirected graph. If G is undirected, then we give an algorithm to compute a shortest ..."
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Cited by 5 (3 self)
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Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of nontrivial cycles in G, using different techniques depending on whether or not G is an undirected graph. If G is undirected, then we give an algorithm to compute a shortest nonseparating cycle in G in 2O(g) n log log n time. Similar algorithms are given to compute a shortest noncontractible or nonnullhomologous cycle in 2O(g+b) n log log n time. Our algorithms for undirected G combine an algorithm of Kutz with known techniques for efficiently enumerating homotopy classes of curves that may be shortest nontrivial cycles. Our main technical contributions in this work arise from assuming G is a directed graph with possibly asymmetric edge weights. For this case, we give an algorithm to compute a shortest noncontractible cycle in G in O((g 3 + g b)n log n) time. In order to achieve this time bound, we use a restriction of the infinite cyclic cover that may be useful in other contexts. We also describe an algorithm to compute a shortest nonnullhomologous cycle in G in O((g 2 + g b)n log n) time, extending a known algorithm of Erickson to compute a shortest nonseparating cycle. In both the undirected and directed cases, our algorithms improve the best time bounds known for many values of g and b. 1
Faster shortest noncontractible cycles in directed surface graphs
 CoRR
"... Let G be a directed graph embedded on a surface of genus g with b boundary cycles. We describe an algorithm to compute the shortest noncontractible cycle in G in O((g 3 + g b)n log n) time. Our algorithm improves the previous best known time bound of (g + b) O(g+b) n log n for all positive g and b. ..."
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Cited by 2 (0 self)
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Let G be a directed graph embedded on a surface of genus g with b boundary cycles. We describe an algorithm to compute the shortest noncontractible cycle in G in O((g 3 + g b)n log n) time. Our algorithm improves the previous best known time bound of (g + b) O(g+b) n log n for all positive g and b. We also describe an algorithm to compute the shortest nonnullhomologous cycle in G in O((g 2 + g b)n log n) time, generalizing a known algorithm to compute the shortest nonseparating cycle.
COMPUTING THE GIRTH OF A PLANAR GRAPH IN LINEAR TIME∗
"... Abstract. The girth of a graph is the minimum weight of all simple cycles of the graph. We study the problem of determining the girth of an nnode unweighted undirected planar graph. The first nontrivial algorithm for the problem, given by Djidjev, runs in O(n5/4 logn) time. Chalermsook, Fakcharoenp ..."
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Abstract. The girth of a graph is the minimum weight of all simple cycles of the graph. We study the problem of determining the girth of an nnode unweighted undirected planar graph. The first nontrivial algorithm for the problem, given by Djidjev, runs in O(n5/4 logn) time. Chalermsook, Fakcharoenphol, and Nanongkai reduced the running time to O(n log2 n). Weimann and Yuster further reduced the running time to O(n logn). In this paper, we solve the problem in O(n) time.
FAST ALGORITHMS FOR SURFACE EMBEDDED GRAPHS VIA HOMOLOGY
, 2013
"... We describe several results on combinatorial optimization problems for graphs where the input comes with an embedding on an orientable surface of small genus. While the specific techniques used differ between problems, all the algorithms we describe share one common feature in that they rely on the ..."
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We describe several results on combinatorial optimization problems for graphs where the input comes with an embedding on an orientable surface of small genus. While the specific techniques used differ between problems, all the algorithms we describe share one common feature in that they rely on the algebraic topology construct of homology. We describe algorithms to compute global minimum cuts and count minimum s, tcuts. We describe new algorithms to compute short cycles that are topologically nontrivial. Finally, we describe ongoing work in designing a new algorithm for computing maximum s, tflows in surface embedded graphs. We begin by describing an algorithm to compute global minimum cuts in edge weighted genus g graphs in gO(g)n log log n time. When the genus is a constant, our algorithm’s running time matches the best time bound known for planar graphs due to La̧cki and Sankowski. In our algorithm, we reduce to the problem of finding a minimum weight separating subgraph in the dual