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18
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 15 (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.
Multiplesource shortest paths in embedded graphs
, 2012
"... Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of ..."
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Cited by 12 (6 self)
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Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)time algorithm of Klein [Multiplesource shortest paths in planar graphs. In Proc. 16th Ann. ACMSIAM Symp. Discrete Algorithms, 2005] for multiplesource shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortestpath tree as its source point moves continuously around the boundary of f. As an application of our algorithm, we describe algorithms to compute a shortest noncontractible or nonseparating cycle in embedded, undirected graphs in O(g² n log n) time.
Finding cycles with topological properties in embedded graphs
, 2010
"... Let G be a graph cellularly embedded on a surface. We consider the problem of determining whether G contains a cycle (i.e. a closed walk without repeated vertices) of a certain topological type. We show that the problem can be answered in linear time when the topological type is one of the following ..."
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Cited by 8 (1 self)
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Let G be a graph cellularly embedded on a surface. We consider the problem of determining whether G contains a cycle (i.e. a closed walk without repeated vertices) of a certain topological type. We show that the problem can be answered in linear time when the topological type is one of the following: contractible, noncontractible, or nonseparating. In either case we obtain the same time complexity if we require the cycle to contain a given vertex. On the other hand, we prove that the problem is NPcomplete when considering separating or splitting cycles. We also show that deciding the existence of a separating or a splitting cycle of length at most k is fixedparameter tractable with respect tok plus the genus of the surface.
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 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
Annotating simplices with a homology basis and its applications
 In Proc. 13th Scandinavian Symp. and Workshop on Algorithm Theory
, 2012
"... Let K be a simplicial complex and g the rank of its pth homology group Hp(K) defined with Z2 coefficients. We show that we can compute a basis H of Hp(K) and annotate each psimplex of K with a binary vector of length g with the following property: the annotations, summed over all psimplices in an ..."
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Cited by 5 (4 self)
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Let K be a simplicial complex and g the rank of its pth homology group Hp(K) defined with Z2 coefficients. We show that we can compute a basis H of Hp(K) and annotate each psimplex of K with a binary vector of length g with the following property: the annotations, summed over all psimplices in any pcycle z, provide the coordinate vector of the homology class [z] in the basis H. The basis and the annotations for all simplices can be computed in O(n ω) time, where n is the size of K and ω < 2.376 is a quantity so that two n × n matrices can be multiplied in O(n ω) time. The precomputation of annotations permits answering queries about the independence or the triviality of pcycles efficiently. Using annotations of edges in 2complexes, we derive better algorithms for computing optimal basis and optimal homologous cycles in 1dimensional homology. Specifically, for computing an optimal basis of H1(K), we improve the time complexity known for the problem from O(n 4) to O(n ω + n 2 g ω−1). Here n denotes the size of the 2skeleton of K and g the rank of H1(K). Computing an optimal cycle homologous to a given 1cycle is NPhard even for surfaces and an algorithm taking 2 O(g) n log n time is known for surfaces. We extend this algorithm to work with arbitrary 2complexes in O(n ω) + 2 O(g) n 2 log n time using annotations. 1
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 5 (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
Combinatorial Optimization of Cycles and Bases
 PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS
"... We survey algorithms and hardness results for two important classes of topology optimization problems: computing minimumweight cycles in a given homotopy or homology class, and computing minimumweight cycle bases for the fundamental group or various homology groups. ..."
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Cited by 4 (0 self)
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We survey algorithms and hardness results for two important classes of topology optimization problems: computing minimumweight cycles in a given homotopy or homology class, and computing minimumweight cycle bases for the fundamental group or various homology groups.
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.
An Efficient Computation of Handle and Tunnel Loops via Reeb Graphs
"... A special family of nontrivial loops on a surface called handle and tunnel loops associates closely to geometric features of “handles” and “tunnels” respectively in a 3D model. The identification of these handle and tunnel loops can benefit a broad range of applications from topology simplificatio ..."
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Cited by 2 (0 self)
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A special family of nontrivial loops on a surface called handle and tunnel loops associates closely to geometric features of “handles” and “tunnels” respectively in a 3D model. The identification of these handle and tunnel loops can benefit a broad range of applications from topology simplification / repair, and surface parameterization, to feature and shape recognition. Many of the existing efficient algorithms for computing nontrivial loops cannot be used to compute these special type of loops. The two algorithms known for computing handle and tunnel loops provably have a serious drawback that they both require a tessellation of the interior and exterior spaces bounded by the surface. Computing such a tessellation of three dimensional space around the surface is a nontrivial task and can be quite expensive. Furthermore, such a tessellation may need to refine the surface mesh, thus causing the undesirable sideeffect of outputting the loops on an altered surface mesh. In this paper, we present an efficient algorithm to compute a basis for handle and tunnel loops without requiring any 3D tessellation. This saves time considerably for large meshes making the algorithm scalable while computing the loops on the original input mesh and not on some refined version of it. We use the concept of the Reeb graph which together with several key theoretical insights on linking number provide an initial set of loops that provably constitute a handle and a tunnel basis. We further develop a novel strategy to tighten these handle and tunnel basis loops to make them geometrically relevant. We demonstrate the efficiency and effectiveness of our algorithm as well as show its robustness against noise, and other anomalies in the input.