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Persistent Triangulations
 UNDER CONSIDERATION FOR PUBLICATION IN J. FUNCTIONAL PROGRAMMING
, 2002
"... Triangulations of a surface are of fundamental importance in computational geometry, computer graphics, and engineering and scientific simulations. Triangulations are ordinarily represented as mutable graph structures for which both adding and traversing edges take constant time per operation. These ..."
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Cited by 7 (2 self)
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Triangulations of a surface are of fundamental importance in computational geometry, computer graphics, and engineering and scientific simulations. Triangulations are ordinarily represented as mutable graph structures for which both adding and traversing edges take constant time per operation. These representations of triangulations make it difficult to support persistence, including “multiple futures”, the ability to use a data structure in several unrelated ways in a given computation; “time travel”, the ability to move freely among versions of a data structure; or parallel computation, the ability to operate concurrently on a data structure without interference. We present a purely functional interface and representation of triangulated surfaces, and more generally of simplicial complexes in higher dimensions. In addition to being persistent in the strongest sense, the interface more closely matches the mathematical definition of triangulations (simplicial complexes) than do interfaces based on mutable representations. The representation, however, comes at the cost of requiring O(lg n) time for traversing or adding triangles (simplices), where n is the number of triangles in the surface. We show both analytically and experimentally that for certain important cases, this extra cost does not seriously affect endtoend running time. Analytically, we present a new randomized algorithm for 3dimensional Convex Hull based on our representations for which the running time matches the Ω(n lg n) lowerbound for the problem. This is achieved by using only O(n) traversals of the surface. Experimentally, we present results for both an implementation of the 3dimensional Convex Hull and for a terrain modeling algorithm, which demonstrate that, although there is some cost to persistence, it seems to be a small constant factor.
A LusternikSchnirelmann theorem for graphs
, 2012
"... We prove the discrete LusternikSchnirelmann theorem tcat(G) ≤ crit(G) for general simple graphs G = (V, E). It relates tcat(G), the minimal number of in G contractible graphs covering G, with crit(G), the minimal number of critical points which an injective function f: V → R can have. Also the cup ..."
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We prove the discrete LusternikSchnirelmann theorem tcat(G) ≤ crit(G) for general simple graphs G = (V, E). It relates tcat(G), the minimal number of in G contractible graphs covering G, with crit(G), the minimal number of critical points which an injective function f: V → R can have. Also the cup length estimate cup(G) ≤ tcat(G) is valid for any finite simple graph. Let cat(G) be the minimal tcat(H) among all H homotopic to G and let cri(G) be the minimal crit(H) among all graphs H homotopic to G, then cup(G) ≤ cat(G) ≤ cri(G) relates three homotopy invariants for graphs: the algebraic cup(G), the topological cat(G) and the analytic cri(G).
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
On critical kernels
, 2005
"... We propose a method for collapsing simplicial complexes in a symmetric manner. For that purpose, we introduce the notions of a simple cell, of an essential face, and the one of a core of a cell. Then, we define the critical kernel of a complex. Our main result is that the critical kernel of a given ..."
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We propose a method for collapsing simplicial complexes in a symmetric manner. For that purpose, we introduce the notions of a simple cell, of an essential face, and the one of a core of a cell. Then, we define the critical kernel of a complex. Our main result is that the critical kernel of a given complex X is a collapse of X. We extend this result by giving a necessary and sufficient condition which characterizes a certain class of subcomplexes of X which contain the critical kernel of X. In particular, any complex which belongs to this class is homotopy equivalent to X. 1
Some properties of topological greyscale watersheds
 ROCS. SPIE VISION GEOMETRY XII, FRANCE
, 2004
"... In this paper, we investigate topological watersheds. 1 For that purpose we introduce a notion of “separation between two points ” of an image. One of our main results is a necessary and sufficient condition for a map G to be a watershed of a map F, this condition is based on the notion of separatio ..."
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Cited by 4 (3 self)
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In this paper, we investigate topological watersheds. 1 For that purpose we introduce a notion of “separation between two points ” of an image. One of our main results is a necessary and sufficient condition for a map G to be a watershed of a map F, this condition is based on the notion of separation. A consequence of the theorem is that there exists a (greedy) polynomial time algorithm to decide whether a map G is a watershed of a map F or not. We also show that, given an arbitrary total order on the minima of a map, it is possible to define a notion of “degree of separation of a minimum ” relative to this order. This leads to another necessary and sufficient condition for a map G to be a watershed of a map F. At last we derive, from our framework, a new definition for the dynamics of a minimum.
Determining edge expansion and other connectivity measures of graphs of bounded genus
 In: Proc. 18th Ann. Europ. Symp. Algorithms, ESA’10
, 2010
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Persistent Homology: An Introduction and a New Text Representation for Natural Language Processing
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Correctness Criteria based on a Homology of Proof Structures in Multiplicative Linear Logic
, 1996
"... Contents 1 Introduction 1 2 Multiplicative linear logic 5 2.1 The calculus : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 Cut elimination : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.3 Strong normalization : : : : : : : : : : : : : : : ..."
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Cited by 2 (0 self)
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Contents 1 Introduction 1 2 Multiplicative linear logic 5 2.1 The calculus : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 Cut elimination : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.3 Strong normalization : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 3 Graphs 15 3.1 Paired directed multigraphs : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 3.2 Constructions on pdm's : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17 3.3 Proof nets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 4 Homology 27 4.1 General theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 4.2 Application on pdm's : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31