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Multiple source shortest paths in a genus g graph
 Proc. 18th Ann. ACMSIAM Symp. Discrete Algorithms
"... We give an O(g2n log n) algorithm to represent the shortest path tree from all the vertices on a single specified face f in a genus g graph. From this representation, any query distance from a vertex in f can be obtained in O(log n) time. The algorithm uses a kinetic data structure, where the source ..."
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Cited by 36 (12 self)
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We give an O(g2n log n) algorithm to represent the shortest path tree from all the vertices on a single specified face f in a genus g graph. From this representation, any query distance from a vertex in f can be obtained in O(log n) time. The algorithm uses a kinetic data structure, where the source of the tree iteratively movesacrossedgesinf. In addition, we give applications using these shortest path trees in order to compute the shortest noncontractible cycle and the shortest nonseparating cycle embedded on an orientable 2manifold in O(g3n log n) time. 1
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 (11 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.
Splitting (complicated) surfaces is hard
 Comput. Geom. Theory Appl
, 2008
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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Cited by 28 (11 self)
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
I/Oefficient batched unionfind and its applications to terrain analysis
 IN PROC. 22ND ANNUAL SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2006
"... Despite extensive study over the last four decades and numerous applications, no I/Oefficient algorithm is known for the unionfind problem. In this paper we present an I/Oefficient algorithm for the batched (offline) version of the unionfind problem. Given any sequence of N union and find opera ..."
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Cited by 24 (9 self)
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Despite extensive study over the last four decades and numerous applications, no I/Oefficient algorithm is known for the unionfind problem. In this paper we present an I/Oefficient algorithm for the batched (offline) version of the unionfind problem. Given any sequence of N union and find operations, where each union operation joins two distinct sets, our algorithm uses O(SORT(N)) = O ( N B log M/B N I/Os, where M is the memory size and B is the disk block size. This bound is asymptotically optimal in the worst case. If there are union operations that join a set with itself, our algorithm uses O(SORT(N) + MST(N)) I/Os, where MST(N) is the number of I/Os needed to compute the minimum spanning tree of a graph with N edges. We also describe a simple and practical O(SORT(N) log ( N M))I/O algorithm for this problem, which we have implemented. We are interested in the unionfind problem because of its applications in terrain analysis. A terrain can be abstracted as a height function defined over R2, and many problems that deal with such functions require a unionfind data structure. With the emergence of modern mapping technologies, huge amount of elevation data is being generated that is too large to fit in memory, thus I/Oefficient algorithms are needed to process this data efficiently. In this paper, we study two terrainanalysis problems that benefit from a unionfind data structure: (i) computing topological persistence and (ii) constructing the contour tree. We give the first O(SORT(N))I/O algorithms for these two problems, assuming that the input terrain is represented as a triangular mesh with N vertices. Finally, we report some preliminary experimental results, showing that our algorithms give orderofmagnitude improvement over previous methods on large data sets that do not fit in memory.
Localized homology
 In Shape Modeling International
, 2007
"... In this paper, we provide the theoretical foundation and an effective algorithm for localizing topological attributes such as tunnels and voids. Unlike previous work that focused on 2manifolds with restricted geometry, our theory is general and localizes arbitrarydimensional attributes in arbitrar ..."
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Cited by 19 (4 self)
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In this paper, we provide the theoretical foundation and an effective algorithm for localizing topological attributes such as tunnels and voids. Unlike previous work that focused on 2manifolds with restricted geometry, our theory is general and localizes arbitrarydimensional attributes in arbitrary spaces. We implement our algorithm to validate our approach in practice. 1
Computing the shortest essential cycle
, 2008
"... An essential cycle on a surface is a simple cycle that cannot be continuously deformed to a point or a single boundary. We describe algorithms to compute the shortest essential cycle in an orientable combinatorial surface in O(n 2 log n) time, or in O(n log n) time when both the genus and number of ..."
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Cited by 12 (4 self)
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An essential cycle on a surface is a simple cycle that cannot be continuously deformed to a point or a single boundary. We describe algorithms to compute the shortest essential cycle in an orientable combinatorial surface in O(n 2 log n) time, or in O(n log n) time when both the genus and number of boundaries are fixed. Our result corrects an error in a paper of Erickson and HarPeled.
Detecting Symmetry in Scalar Fields Using Augmented Extremum Graphs
"... Fig. 1. Robust scalar field symmetry identification algorithm detects symmetry even in the presence of significant noise in the electron microscopy data of the Rubisco RbcL8RbcX28 complex (EMDB 1654). (left) Volume rendering shows symmetry and noise in the data. (center) A set of seed cells is cho ..."
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Cited by 4 (1 self)
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Fig. 1. Robust scalar field symmetry identification algorithm detects symmetry even in the presence of significant noise in the electron microscopy data of the Rubisco RbcL8RbcX28 complex (EMDB 1654). (left) Volume rendering shows symmetry and noise in the data. (center) A set of seed cells is chosen as source vertices for traversing the augmented extremum graph of the data. During the traversal, the seed cells merge together to form four symmetric superseeds. Seed cells that belong to a common superseed are shown with the same color. (right) The initial estimate of symmetry is expanded in a region growing stage to identify the symmetric regions. A symmetryaware transfer function highlights the 4fold rotational symmetry detected in the Rubisco complex. Abstract—Visualizing symmetric patterns in the data often helps the domain scientists make important observations and gain insights about the underlying experiment. Detecting symmetry in scalar fields is a nascent area of research and existing methods that detect symmetry are either not robust in the presence of noise or computationally costly. We propose a data structure called the augmented extremum graph and use it to design a novel symmetry detection method based on robust estimation of distances. The augmented extremum graph captures both topological and geometric information of the scalar field and enables robust and computationally efficient detection of symmetry. We apply the proposed method to detect symmetries in cryoelectron microscopy datasets and the experiments demonstrate that the algorithm is capable of detecting symmetry even in the presence of significant noise. We describe novel applications that use the detected symmetry to enhance visualization of scalar field data and facilitate their exploration. Index Terms—Scalar field visualization, extremum graph, Morse decomposition, symmetry detection, data exploration. 1
Testing contractibility in planar Rips complexes
 In Proc. Symp. on Comp. Geom. (SoCG) 2008
"... The (Vietoris)Rips complex of a discrete pointset P is an abstract simplicial complex in which a subset of P defines a simplex if and only if the diameter of that subset is at most 1. We describe an efficient algorithm to determine whether a given cycle in a planar Rips complex is contractible. Ou ..."
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The (Vietoris)Rips complex of a discrete pointset P is an abstract simplicial complex in which a subset of P defines a simplex if and only if the diameter of that subset is at most 1. We describe an efficient algorithm to determine whether a given cycle in a planar Rips complex is contractible. Our algorithm requires O(m log n) time to preprocess a set of n points in the plane in which m pairs have distance at most 1; after preprocessing, deciding whether a cycle of k Rips edges is contractible requires O(k) time. We also describe an algorithm to compute the shortest noncontractible cycle in a planar Rips complex in O(n 2 log n + mn) time.
Alpha Shape Topology of the Cosmic Web
"... Abstract—We study the topology of the Megaparsec Cosmic Web on the basis of the Alpha Shapes of the galaxy distribution. The simplicial complexes of the alpha shapes are used to determine the set of Betti numbers (βk, k = 1,..., D), which represent a complete characterization of the topology of a ma ..."
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Abstract—We study the topology of the Megaparsec Cosmic Web on the basis of the Alpha Shapes of the galaxy distribution. The simplicial complexes of the alpha shapes are used to determine the set of Betti numbers (βk, k = 1,..., D), which represent a complete characterization of the topology of a manifold. This forms a useful extension of the geometry and topology of the galaxy distribution by Minkowski functionals, of which three specify the geometrical structure of surfaces and one, the Euler characteristic, represents a key aspect of its topology. In order to develop an intuitive understanding for the relation between Betti numbers and the running α parameter of the alpha shapes, and thus in how far they may discriminate between different topologies, we study them within the context of simple heuristic Voronoi clustering models. These may be tuned to consist of a few or even only one specific morphological element of the Cosmic Web, ie. clusters, filaments or sheets. KeywordsCosmology: theory largescale structure of Uni