共查询到20条相似文献,搜索用时 15 毫秒
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Kree Cole-McLaughlin Herbert Edelsbrunner John Harer Vijay Natarajan Valerio Pascucci 《Discrete and Computational Geometry》2004,32(2):231-244
Given a Morse function f over a 2-manifold with or without boundary,
the Reeb graph is obtained by contracting the connected components
of the level sets to points.
We prove tight upper and lower bounds on the
number of loops in the Reeb graph
that depend on the genus, the number of boundary components,
and whether or not the 2-manifold is orientable.
We also give an algorithm that constructs the Reeb graph in time
O(n log n), where n is the number of edges in the
triangulation used to represent the 2-manifold and the Morse function. 相似文献
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Persistence approximation property was introduced by Hervé Oyono-Oyono and Guoliang Yu. This property provides a geometric obstruction to Baum-Connes conjecture. In this paper, the authors mainly discuss the persistence approximation property for maximal Roe algebras. They show that persistence approximation property of maximal Roe algebras follows from maximal coarse Baum-Connes conjecture. In particular, let X be a discrete metric space with bounded geometry, assume that X admits a fibred coar... 相似文献
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The problem to find a 4-edge-coloring of a 3-regular graph is solvable in polynomial time but an analogous problem for 3-edge-coloring is NP-hard. We study complexity of approximation algorithms for invariants measuring how far is a 3-regular graph from having a 3-edge-coloring. We prove that it is an NP-hard problem to approximate such invariants by a power function with exponent smaller than 1. 相似文献
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We prove that the genus of the boundary of a digital image is precisely half of the sum of the cycle ranks of three particular
graphs: the "foreground graph" and "background graph," which capture topological information about the digital image and its
complement, respectively, and the Reeb graph, relative to the natural height function, associated with the digital image's
boundary. We prove several additional results, including a characterization of when the cycle rank of the Reeb graph fails
to equal the genus of the digital image's boundary (which can happen by virtue of the failure of the natural height function
on the boundary of the digital image to be a Morse function). 相似文献
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Received: August 22, 1996 相似文献
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《Discrete Mathematics》2020,343(2):111658
A well known result in the analysis of finite metric spaces due to Gromov says that given any metric space there exists a tree metric on such that is bounded above by twice . Here is the hyperbolicity of , a quantity that measures the treeness of 4-tuples of points in . This bound is known to be asymptotically tight.We improve this bound by restricting ourselves to metric spaces arising from filtered posets. By doing so we are able to replace the cardinality appearing in Gromov’s bound by a certain poset theoretic invariant which can be much smaller thus significantly improving the approximation bound.The setting of metric spaces arising from posets is rich: For example, every finite metric graph can be induced from a filtered poset. Since every finite metric space can be isometrically embedded into a finite metric graph, our ideas are applicable to finite metric spaces as well.At the core of our results lies the adaptation of the Reeb graph and Reeb tree constructions and the concept of hyperbolicity to the setting of posets, which we use to formulate and prove a tree approximation result for any filtered poset. 相似文献
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V. Chepoi F. F. Dragan I. Newman Y. Rabinovich Y. Vaxès 《Discrete and Computational Geometry》2012,47(1):187-214
In this paper, we present a simple factor 6 algorithm for approximating the optimal multiplicative distortion of embedding
a graph metric into a tree metric (thus improving and simplifying the factor 100 and 27 algorithms of Bǎdoiu et al. (Proceedings
of the eighteenth annual ACM–SIAM symposium on discrete algorithms (SODA 2007), pp. 512–521, 2007) and Bǎdoiu et al. (Proceedings of the 11th international workshop on approximation algorithms for combinatorial optimization
problems (APPROX 2008), Springer, Berlin, pp. 21–34, 2008)). We also present a constant factor algorithm for approximating the optimal distortion of embedding a graph metric into
an outerplanar metric. For this, we introduce a general notion of a metric relaxed minor and show that if G contains an α-metric relaxed H-minor, then the distortion of any embedding of G into any metric induced by a H-minor free graph is ≥α. Then, for H=K
2,3, we present an algorithm which either finds an α-relaxed minor, or produces an O(α)-embedding into an outerplanar metric. 相似文献
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Frol Zapolsky 《Israel Journal of Mathematics》2012,188(1):111-121
This note deals with quasi-states on the two-dimensional torus. Quasistates are certain quasi-linear functionals (introduced
by Aarnes) on the space of continuous functions. Grubb constructed a quasi-state on the torus, which is invariant under the
group of area-preserving diffeomorphisms, and which moreover vanishes on functions having support in an open disk. Knudsen
asserted the uniqueness of such a quasi-state; for the sake of completeness, we provide a proof. We calculate the value of
Grubb’s quasi-state on Morse functions with distinct critical values via their Reeb graphs. The resulting formula coincides
with the one obtained by Py in his work on quasi-morphisms on the group of area-preserving diffeomorphisms of the torus. Included
is a short introduction to the link between quasi-states and quasi-morphisms in symplectic geometry. 相似文献
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William P. Thurston 《Topology》1974,13(4):347-352
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Let φ be any flow on T n obtained as the suspension of a smooth diffeomorphism of \({T^{n-1}}\) , and let \({\mathcal {A}}\) be any compact invariant set of φ. We realize \({(\mathcal{A}, \varphi|_{\mathcal{A}})}\) up to reparametrization as an invariant set of the Reeb flow of a contact form on \({\mathbb{R}^{2n+1}}\) equal to the standard contact form outside a compact set and defining the standard contact structure on all of \({\mathbb{R}^{2n+1}}\) . This uses the method from Geiges, Röttgen, and Zehmisch. 相似文献
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Bauer Ulrich Landi Claudia Mémoli Facundo 《Foundations of Computational Mathematics》2021,21(5):1441-1464
Foundations of Computational Mathematics - We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are... 相似文献
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