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DavenportSchinzel Sequences and Their Geometric Applications
, 1998
"... An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \ ..."
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Cited by 439 (105 self)
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of geometric problems can be formulated in terms of lower envelopes. A nearlinear bound on the maximum length of DavenportSchinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
The geometry of graphs and some of its algorithmic applications
 COMBINATORICA
, 1995
"... In this paper we explore some implications of viewing graphs as geometric objects. This approach offers a new perspective on a number of graphtheoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that res ..."
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Cited by 524 (19 self)
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their geometric images. In this paper we develop efficient algorithms for embedding graphs lowdimensionally with a small distortion. Further algorithmic applications include: 0 A simple, unified approach to a number of problems on multicommodity flows, including the LeightonRae Theorem [29] and some of its ex
THREE GEOMETRIC APPLICATIONS OF QUANDLE HOMOLOGY
, 2008
"... In this paper we describe three geometric applications of quandle homology. We show that it gives obstructions to tangle embeddings, provides the lower bound for the 4move distance between links, and can be used in determining periodicity of links. ..."
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Cited by 2 (0 self)
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In this paper we describe three geometric applications of quandle homology. We show that it gives obstructions to tangle embeddings, provides the lower bound for the 4move distance between links, and can be used in determining periodicity of links.
GEOMETRIC APPLICATIONS OF WASSERSTEIN DISTANCE
"... Wasserstein distance and optimal transportation The lecture will be devoted to the Wassertein distance of Borel probability measures, which arises from the optimal transportation theory [4] [5]. A number of examples will illustrate the nature of this metric, which is defined on the space of all Bore ..."
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Wasserstein distance and optimal transportation The lecture will be devoted to the Wassertein distance of Borel probability measures, which arises from the optimal transportation theory [4] [5]. A number of examples will illustrate the nature of this metric, which is defined on the space of all Borel probability measures. Weak ⋆ Metrization Theorem will be presented. Lecture 2 Curvature of metric measure spaces I The second lecture will present the notion of the lower curvature bound [3] for a metric measure spaces. Some basic concepts of metric geometry (length spaces, geodesics, geodesic spaces) will be recalled. A kind of a generalization of Wasserstein distance, namely the WassersteinGromov distance of metric measure spaces and the relative entropy of measures will be the main definitions that will be presented. Lecture 3 Curvature of metric measure spaces II This lecture will be the continuation of the previous one. The lower curvature
Geometric Applications of Posets
, 1998
"... We show the power of posets in computational geometry by solving several problems posed on a set S of n points in the plane: (1) find the n \Gamma k \Gamma 1 rectilinear farthest neighbors (or, equivalently, k nearest neighbors) to every point of S (extendable to higher dimensions), (2) enumerate th ..."
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We show the power of posets in computational geometry by solving several problems posed on a set S of n points in the plane: (1) find the n \Gamma k \Gamma 1 rectilinear farthest neighbors (or, equivalently, k nearest neighbors) to every point of S (extendable to higher dimensions), (2) enumerate the k largest (smallest) rectilinear distances in decreasing (increasing) order among the points of S, (3) given a distance ffi ? 0, report all the pairs of points that belong to S and are of rectilinear distance ffi or more (less), covering k n 2 points of S by rectilinear (4) and circular (5) concentric rings, and (6) given a number k n 2 decide whether a query rectangle contains k points or less.
Laplacian Eigenmaps for Dimensionality Reduction and Data Representation
, 2003
"... One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a lowdimensional manifold embedded in a highdimensional space. Drawing on the correspondenc ..."
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Cited by 1226 (15 self)
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on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for representing the highdimensional data. The algorithm provides a computationally efficient approach to nonlinear dimensionality
No Free Lunch Theorems for Optimization
, 1997
"... A framework is developed to explore the connection between effective optimization algorithms and the problems they are solving. A number of “no free lunch ” (NFL) theorems are presented which establish that for any algorithm, any elevated performance over one class of problems is offset by performan ..."
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Cited by 961 (10 self)
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by performance over another class. These theorems result in a geometric interpretation of what it means for an algorithm to be well suited to an optimization problem. Applications of the NFL theorems to informationtheoretic aspects of optimization and benchmark measures of performance are also presented. Other
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