character degrees and nilpotence class of finite p groups an approach via pro p groups let s be a finite set of powers of p containing 1 it is known that for some choices of s if p is a finite p group whose set of character degrees is s then the nilpotence class of p is bounded by some integer that depends on s while for some other choices of s such an integer does not exist the sets of the first type are called class bounding sets the problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set s is class bounding if and only if p s in this article we provide a new approach to this problem our main result shows the relevance of certain p adic space groups in this problem with its help we are able to prove some results that provide new class bounding sets we also show that there exist non class bounding sets s such that p s

centrality of trees for capacitated k k center we consider the capacitated k k center problem in this problem we are given a finite set of locations in a metric space and each location has an associated non negative integer capacity the goal is to choose open k k locations called centers and assign each location to an open center to minimize the maximum over all locations of the distance of the location to its assigned center the number of locations assigned to a center cannot exceed the center s capacity the uncapacitated k k center problem has a simple tight 2 2 approximation from the 80 s in contrast the first constant factor approximation for the capacitated problem was obtained only recently by cygan hajiaghayi and khuller who gave an intricate lp rounding algorithm that achieves an approximation guarantee in the hundreds in this paper we give a simple algorithm with a clean analysis and prove an approximation guarantee of 9 9 it uses the standard lp relaxation and comes close to settling the integrality gap after necessary preprocessing which is narrowed down to either 7 8 7 8 or 9 9 the algorithm proceeds by first reducing to special tree instances and then uses our best possible algorithm to solve such instances our concept of tree instances is versatile and applies to natural variants of the capacitated k k center problem for which we also obtain improved algorithms finally we give evidence to show that more powerful preprocessing could lead to better algorithms by giving an approximation algorithm that beats the integrality gap for instances where all non zero capacities are the same

k center problems with minimum coverage in this work we study an extension of the k center facility location problem where centers are required to service a minimum of clients this problem is motivated by requirements to balance the workload of centers while allowing each center to cater to a spread of clients we study three variants of this problem all of which are shown to be n p hard in approximation hardness and approximation algorithms with factors equal or close to the best lower bounds are provided generalizations including vertex costs and vertex weights are also studied

decentralized clustering by finding loose and distributed density cores abstract centroid based clustering approaches fail to recognize extremely complex patterns that are non isotropic we analyze the underlying causes and find some inherent flaws in these approaches including shape loss false distances and false peaks which typically cause centroid based approaches to fail when applied to complex patterns as an alternative to current methods we propose a hybrid decentralized approach named dcore which is based on finding density cores instead of centroids to overcome these flaws the underlying idea is that we consider each cluster to have a shrunken density core that roughly retains the shape of the cluster each such core consists of a set of loosely connected local density peaks of higher density than their surroundings borders edges and outliers are distributed around the outsides of these cores in a hierarchical structure experiments demonstrate that the promise of dcore lies in its power to recognize extremely complex patterns and its high performance in real applications for example image segmentation and face clustering regardless of the dimensionality of the space in which the data are embedded

the ordered k median problem surrogate models and approximation algorithms in the last two decades a steady stream of research has been devoted to studying various computational aspects of the ordered k median problem which subsumes traditional facility location problems such as median center p centrum etc through a unified modeling approach given a finite metric space the objective is to locate k facilities in order to minimize the ordered median cost function in its general form this function penalizes the coverage distance of each vertex by a multiplicative weight depending on its ranking or percentile in the ordered list of all coverage distances while antecedent literature has focused on mathematical properties of ordered median functions integer programming methods various heuristics and special cases this problem was not studied thus far through the lens of approximation algorithms in particular even on simple network topologies such as trees or line graphs obtaining non trivial approximation guarantees is an open question the main contribution of this paper is to devise the first provably good approximation algorithms for the ordered k median problem we develop a novel approach that relies primarily on a surrogate model where the ordered median function is replaced by a simplified ranking invariant functional form via efficient enumeration surprisingly while this surrogate model is varomega n varomega 1 hard to approximate on general metrics we obtain an o log n approximation for our original problem by employing local search methods on a smooth variant of the surrogate function in addition an improved guarantee of 2 epsilon is obtained on tree metrics by optimally solving the surrogate model through dynamic programming finally we show that the latter optimality gap is tight up to an o epsilon term

a gromov hausdorff framework with diffusion geometry for topologically robust non rigid shape matching in this paper the problem of non rigid shape recognition is studied from the perspective of metric geometry in particular we explore the applicability of diffusion distances within the gromov hausdorff framework while the traditionally used geodesic distance exploits the shortest path between points on the surface the diffusion distance averages all paths connecting the points the diffusion distance constitutes an intrinsic metric which is robust in particular to topological changes such changes in the form of shortcuts holes and missing data may be a result of natural non rigid deformations as well as acquisition and representation noise due to inaccurate surface construction the presentation of the proposed framework is complemented with examples demonstrating that in addition to the relatively low complexity involved in the computation of the diffusion distances between surface points its recognition and matching performances favorably compare to the classical geodesic distances in the presence of topological changes between the non rigid shapes

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

progressive scattered data filtering given a finite point set z rd the covering radius of a nonempty subset x z is the minimum distance r x z such that every point in z is at a distance of at most r x z from some point in x this paper concerns the construction of a sequence of subsets of decreasing sizes such that their covering radii are small to this end a method for progressive data reduction referred to as scattered data filtering is proposed the resulting scheme is a composition of greedy thinning a recursive point removal strategy and exchange a postprocessing local optimization procedure the paper proves adaptive a priori lower bounds on the minimal covering radii which allows us to control for any current subset the deviation of its covering radius from the optimal value at run time important computational aspects of greedy thinning and exchange are discussed the good performance of the proposed filtering scheme is finally shown by numerical examples

a new assignment rule to improve seed points algorithms for the continuous k center problem abstract due to its complexity the k center problem cannot be solved exactly when the number of points is high enough so heuristic algorithms must be used to find good solutions for this location allocation problem the allocation part of the heuristics given in the literature consists of assigning each given point to its nearest center in this paper we propose a new assignment rule nar from which a new class of seed points algorithms is obtained for the k center problem in r n the new class differs from the usual class of seed points algorithms in the way the points are assigned to each seed point different methods to generate seed points are shown from which different algorithms are given by using the two assignment rules the classical and the new most of these algorithms have linear complexity in the number of points and generate solutions with objective values within two times the optimal value and all of them run very quickly and can be used for any metric it is shown by computational experience that the running times and the quality of the solutions generated are notably improved by using the nar

lexicographic local search and the p center problem abstract we introduce a local search strategy that suits combinatorial optimization problems with a min max or max min objective according to this approach solutions are compared lexicographically rather then by their worst coordinate we apply this approach to the p center problem

a best possible heuristic for the k center problem in this paper we present a 2 approximation algorithm for the k center problem with triangle inequality this result is best possible since for any 2 the existence of approximation algorithm would imply that p np it should be noted that no approximation algorithm for any constant has been reported to date linear programming duality theory provides interesting insight to the problem and enables us to derive in o e log e time a solution with value no more than twice the k center optimal value a by product of the analysis is an o e algorithm that identifies a dominating set in g2 the square of a graph g the size of which is no larger than the size of the minimum dominating set in the graph g the key combinatorial object used is called a strong stable set and we prove the np completeness of the corresponding decision problem

sharp constants in the hardy rellich inequalities optimal constants are found in hardy rellich inequalities containing derivatives of arbitrary not necessarily integer order l some new inequalities of this type are also obtained

a 2 approximation algorithm and beyond for the minimum diameter k steiner forest problem abstract given an edge weighted undirected graph g v e w and a subset t v of p terminals a k steiner forest spanning all the terminals in t includes k branches where every branch is a steiner tree the diameter of a k steiner forest is referred to as the maximum distance between two terminals of a branch this paper studies the minimum diameter k steiner forest problem mdksfp and establishes the relationship between mdksfp and the absolute k steiner center problem akscp we first obtain a 2 factor dual approximation algorithm for akscp and then achieve a 2 approximation algorithm for mdksfp based on the 2 approximation to akscp furthermore we develop an improved 2 approximation algorithm for mdksfp where 1 in general by perturbing the sites of facilities and re clustering the terminals

cpcq contrast pattern based clustering quality index for categorical data clustering validation is concerned with assessing the quality of clustering solutions since clustering is unsupervised and highly explorative clustering validation has been an important and long standing research problem existing validity measures including entropy based and distance based indices have significant shortcomings indeed for many datasets from the uci repository they fail to recognize that the expert determined classes are the best clusters and they frequently give preference to clusterings with larger number of clusters their weakness reflects their inability to accurately capture intra cluster coherence and inter cluster separation this paper proposes a novel contrast pattern based clustering quality index cpcq for categorical data by utilizing the quality and diversity of the contrast patterns which contrast the clusters in given clusterings high quality contrast patterns can serve to characterize the clusters and discriminate one cluster against the others the cpcq index is based on the rationale that a high quality clustering should have many diversified high quality contrast patterns among its clusters the quality of individual contrast patterns is defined in terms of their length support and the length of their corresponding closed pattern the quality measure concerning many diversified contrast patterns is defined in terms of the quality and diversity of some selected groups of contrast patterns with minimal overlap among contrast patterns and groups in terms of items and matching transactions experiments show that the cpcq index 1 does not require a user to provide a distance function 2 does not give inappropriate preference to larger number of clusters 3 can recognize that expert determined classes are the best clusters for many datasets from the uci repository

robust submodular observation selection in many applications one has to actively select among a set of expensive observations before making an informed decision for example in environmental monitoring we want to select locations to measure in order to most effectively predict spatial phenomena often we want to select observations which are robust against a number of possible objective functions examples include minimizing the maximum posterior variance in gaussian process regression robust experimental design and sensor placement for outbreak detection in this paper we present the submodular saturation algorithm a simple and efficient algorithm with strong theoretical approximation guarantees for cases where the possible objective functions exhibit submodularity an intuitive diminishing returns property moreover we prove that better approximation algorithms do not exist unless np complete problems admit efficient algorithms we show how our algorithm can be extended to handle complex cost functions incorporating non unit observation cost or communication and path costs we also show how the algorithm can be used to near optimally trade off expected case e g the mean square prediction error in gaussian process regression and worst case e g maximum predictive variance performance we show that many important machine learning problems fit our robust submodular observation selection formalism and provide extensive empirical evaluation on several real world problems for gaussian process regression our algorithm compares favorably with state of the art heuristics described in the geostatistics literature while being simpler faster and providing theoretical guarantees for robust experimental design our algorithm performs favorably compared to sdp based algorithms c 2008 andreas krause h brendan mcmahan carlos guestrin and anupam gupta krause mcmahan guestrin and gupta

spectral generalized multi dimensional scaling multidimensional scaling mds is a family of methods that embed a given set of points into a simple usually flat domain the points are assumed to be sampled from some metric space and the mapping attempts to preserve the distances between each pair of points in the set distances in the target space can be computed analytically in this setting generalized mds is an extension that allows mapping one metric space into another that is mds into target spaces in which distances are evaluated numerically rather than analytically here we propose an efficient approach for computing such mappings between surfaces based on their natural spectral decomposition where the surfaces are treated as sampled metric spaces the resulting spectral gmds procedure enables efficient embedding by incorporating smoothness of the metric structure into the problem thereby substantially reducing the complexity involved in its solution while practically overcoming its non convex nature the method is compared to existing techniques that compute dense correspondence between shapes numerical experiments of the proposed method demonstrate its efficiency and accuracy compared to state of the art approaches especially when isometry invariance is a dominant property

a genetic algorithm using hyper quadtrees for low dimensional k means clustering the k means algorithm is widely used for clustering because of its computational efficiency given n points in d dimensional space and the number of desired clusters k k means seeks a set of k cluster centers so as to minimize the sum of the squared euclidean distance between each point and its nearest cluster center however the algorithm is very sensitive to the initial selection of centers and is likely to converge to partitions that are significantly inferior to the global optimum we present a genetic algorithm ga for evolving centers in the k means algorithm that simultaneously identifies good partitions for a range of values around a specified k the set of centers is represented using a hyper quadtree constructed on the data this representation is exploited in our ga to generate an initial population of good centers and to support a novel crossover operation that selectively passes good subsets of neighboring centers from parents to offspring by swapping subtrees experimental results indicate that our ga finds the global optimum for data sets with known optima and finds good solutions for large simulated data sets

the effects of triangle inequality on the vehicle routing problem abstract routing problems often utilize experimental networks to represent real world scenarios however most ignore the inclusion of triangle inequality violations a phenomenon resulting from delays or rounding errors within a network this work evaluates the effect of both frequency the number of violations and severity the degree of intensity of a violation of triangle inequality and evaluates both solution quality and solution time based on simulated annealing ant colony optimization and savings algorithm methods findings indicate that while both frequency and severity degrade solution quality increased levels of frequency and severity together result in significant adverse affects to solution quality solution time however is not impacted by the presence of triangle inequality violations within the network this information should encourage practitioners to identify delays and maintain the presence of triangle inequality violations in a network to ensure accuracy of solution quality

topology invariant similarity of nonrigid shapes this paper explores the problem of similarity criteria between nonrigid shapes broadly speaking such criteria are divided into intrinsic and extrinsic the first referring to the metric structure of the object and the latter to how it is laid out in the euclidean space both criteria have their advantages and disadvantages extrinsic similarity is sensitive to nonrigid deformations while intrinsic similarity is sensitive to topological noise in this paper we approach the problem from the perspective of metric geometry we show that by unifying the extrinsic and intrinsic similarity criteria it is possible to obtain a stronger topology invariant similarity suitable for comparing deformed shapes with different topology we construct this new joint criterion as a tradeoff between the extrinsic and intrinsic similarity and use it as a set valued distance numerical results demonstrate the efficiency of our approach in cases where using either extrinsic or intrinsic criteria alone would fail

the hardy inequality and nonlinear parabolic equations on carnot groups abstract in this paper we shall investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equation u t g p u v x u p 1 in 0 t 1 p 2 u x 0 u 0 x 0 in u x t 0 on 0 t where g p is the p sub laplacian on a carnot group g and v l loc 1

nonlinear dimensionality reduction by topologically constrained isometric embedding many manifold learning procedures try to embed a given feature data into a flat space of low dimensionality while preserving as much as possible the metric in the natural feature space the embedding process usually relies on distances between neighboring features mainly since distances between features that are far apart from each other often provide an unreliable estimation of the true distance on the feature manifold due to its non convexity distortions resulting from using long geodesics indiscriminately lead to a known limitation of the isomap algorithm when used to map non convex manifolds presented is a framework for nonlinear dimensionality reduction that uses both local and global distances in order to learn the intrinsic geometry of flat manifolds with boundaries the resulting algorithm filters out potentially problematic distances between distant feature points based on the properties of the geodesics connecting those points and their relative distance to the boundary of the feature manifold thus avoiding an inherent limitation of the isomap algorithm since the proposed algorithm matches non local structures it is robust to strong noise we show experimental results demonstrating the advantages of the proposed approach over conventional dimensionality reduction techniques both global and local in nature

multimodal manifold analysis by simultaneous diagonalization of laplacians we construct an extension of spectral and diffusion geometry to multiple modalities through simultaneous diagonalization of laplacian matrices this naturally extends classical data analysis tools based on spectral geometry such as diffusion maps and spectral clustering we provide several synthetic and real examples of manifold learning object classification and clustering showing that the joint spectral geometry better captures the inherent structure of multi modal data we also show the relation of many previous approaches for multimodal manifold analysis to our framework

full and partial symmetries of non rigid shapes symmetry and self similarity are the cornerstone of nature exhibiting themselves through the shapes of natural creations and ubiquitous laws of physics since many natural objects are symmetric the absence of symmetry can often be an indication of some anomaly or abnormal behavior therefore detection of asymmetries is important in numerous practical applications including crystallography medical imaging and face recognition to mention a few conversely the assumption of underlying shape symmetry can facilitate solutions to many problems in shape reconstruction and analysis traditionally symmetries are described as extrinsic geometric properties of the shape while being adequate for rigid shapes such a description is inappropriate for non rigid ones extrinsic symmetry can be broken as a result of shape deformations while its intrinsic symmetry is preserved in this paper we present a generalization of symmetries for non rigid shapes and a numerical framework for their analysis addressing the problems of full and partial exact and approximate symmetry detection and classification

partial similarity of objects or how to compare a centaur to a horse similarity is one of the most important abstract concepts in human perception of the world in computer vision numerous applications deal with comparing objects observed in a scene with some a priori known patterns often it happens that while two objects are not similar they have large similar parts that is they are partially similar here we present a novel approach to quantify partial similarity using the notion of pareto optimality we exemplify our approach on the problems of recognizing non rigid geometric objects images and analyzing text sequences

on the continuous fermat weber problem we give the firstexact algorithmic study of facility location problems that deal with finding a median for acontinuum of demand points in particular we consider versions of the continuousk median fermat weber problem where the goal is to select one or more center points that minimize the average distance to a set of points in a demandregion in such problems the average is computed as an integral over the relevant region versus the usual discrete sum of distances the resulting facility location problems are inherently geometric requiring analysis techniques of computational geometry we provide polynomial time algorithms for various versions of thel1 1 median fermat weber problem we also consider the multiple center version of thel1k median problem which we prove is np hard for largek

a semi supervised approximate spectral clustering algorithm based on hmrf model abstract before clustering we usually have some background knowledge about the data structure pairwise constraints are commonly used background knowledge for graph partition problems pairwise constraints can be naturally added to the graph edge this paper integrates pairwise constraints into the objective function of graph cuts and derive the semi supervised approximate spectral clustering based on hidden markov random fields hmrf this algorithm utilize the mathematical connection between hmrf semi supervised clustering and approximate weighted kernel k means the approximate weighted kernel k means is used to calculate the optimal clustering results of hmrf spectral clustering the effectiveness of the proposed algorithm is verified on several benchmark data sets experiments show that adding more pairwise constraints will help improve the clustering performance our method has advantages for the challenging clustering tasks of large scale nonlinear data because of the high efficiency and less memory consumption

finite solvable groups with at most two nonlinear irreducible characters of each degree abstract the aim of this paper is to investigate the finite solvable groups with at most two nonlinear irreducible characters of each degree

k center and k median problems in graded distances graded distances are generalizations of the euclidean distance on points in r1 they have been used in the study of special cases of np hard problems in this paper we study the k center and k median problems with graded distance matrices we first prove that the k center problem is polynomial time solvable when the distance matrix is graded up the rows or graded down the rows we then prove that the k median problem is np complete when the distance matrix is graded up the rows or graded down the rows an easy special case of the k median problem with graded distance matrices is also discussed

fault tolerant k center problems the basic k center problem is a fundamental facility location problem where we are asked to locate k facilities in a graph and to assign vertices to facilities so as to minimize the maximum distance from a vertex to the facility to which it is assigned this problem is known to be np hard and several optimal approximation algorithms that achieve an approximation factor of 2 have been developed for it we focus our attention on a generalization of this problem where each vertex is required to have a set of k centers close to it in particular we study two different versions of this problem in the first version each vertex is required to have at least centers close to it in the second version each vertex that does not have a center placed on it is required to have at least centers close to it for both these versions we are able to provide polynomial time approximation algorithms that achieve constant approximation factors for any for the first version we give an algorithm that achieves an approximation factor of 3 for any and achieves an approximation factor of 2 for 4 for the second version we provide algorithms with approximation factors of 2 for any the best possible approximation factor for even the basic k center problem is 2 assuming p np in addition we give a polynomial time approximation algorithm for a generalization of the k supplier problem where a subset of at most k supplier nodes must be selected as centers so that every demand node has at least centers close to it for this version our approximation factor is 3 the best possible approximation factor for even the basic k supplier problem is 3 assuming p np

asymmetry in k center variants this paper explores three concepts the k center problem some of its variants and asymmetry the k center problem is fundamental in location theory variants of k center may more accurately model real life problems than the original formulation asymmetry is a significant impediment to approximation in many graph problems such as k center facility location k median and the tsp we give an o log n approximation algorithm for the asymmetric weighted k center problem here the vertices have weights and we are given a total budget for opening centers in the p neighbor variant each vertex must have p unweighted centers nearby we give an o log k bicriteria algorithm using 2k centers for small p finally we show the following three versions of the asymmetric k center problem to be inapproximable priority k center k supplier and outliers with forbidden centers

a link based approach to the cluster ensemble problem cluster ensembles have recently emerged as a powerful alternative to standard cluster analysis aggregating several input data clusterings to generate a single output clustering with improved robustness and stability from the early work these techniques held great promise however most of them generate the final solution based on incomplete information of a cluster ensemble the underlying ensemble information matrix reflects only cluster data point relations while those among clusters are generally overlooked this paper presents a new link based approach to improve the conventional matrix it achieves this using the similarity between clusters that are estimated from a link network model of the ensemble in particular three new link based algorithms are proposed for the underlying similarity assessment the final clustering result is generated from the refined matrix using two different consensus functions of feature based and graph based partitioning this approach is the first to address and explicitly employ the relationship between input partitions which has not been emphasized by recent studies of matrix refinement the effectiveness of the link based approach is empirically demonstrated over 10 data sets synthetic and real and three benchmark evaluation measures the results suggest the new approach is able to efficiently extract information embedded in the input clusterings and regularly illustrate higher clustering quality in comparison to several state of the art techniques

fitting heights of odd order groups with few character degrees abstract if g is any finite solvable group having a normal sylow 2 subgroup in particular if g is odd and satisfying cd g 5 where cd g is the set of ordinary irreducible character degrees of g we show that the fitting height of g does not exceed cd g 2 in case cd g 5 this upper bound on the fitting height of g is best possible

a plant location guide for the unsure approximation algorithms for min max location problems this paper studies an extension of the k median problem under uncertain demand we are given an n vertex metric space v d and m client sets documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document s i subseteq v i 1 m end document the goal is to open a set of k facilities f such that the worst case connection cost over all the client sets is minimized i e documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document min f subseteq v f k max i in m left sum j in s i d

algorithm to find a maximum 2 packing set in a cactus abstract in this paper we address the problem of finding a maximum 2 packing set in a cactus let g v g e g be an undirected connected graph then a subset s v g is a 2 packing set if for every pair of vertices u v s the shortest path between them is at least three edges long this type of sets results in a wide range of applications such as the study of molecules network modeling allocation of facilities and frequency assignment the cactus is a planar graph such that any edge belongs to at most one cycle hitherto to the best of the authors knowledge no algorithm finds a maximum 2 packing set in a cactus this problem is np hard in arbitrary graphs our approach to solving this problem was first finding a maximum 2 packing set in a unicyclic graph then we adapted this algorithm to run in a cactus the time complexity of the resulting algorithm is o n 2 where n v g

spectral multidimensional scaling an important tool in information analysis is dimensionality reduction there are various approaches for large data simplification by scaling its dimensions down that play a significant role in recognition and classification tasks the efficiency of dimension reduction tools is measured in terms of memory and computational complexity which are usually a function of the number of the given data points sparse local operators that involve substantially less than quadratic complexity at one end and faithful multiscale models with quadratic cost at the other end make the design of dimension reduction procedure a delicate balance between modeling accuracy and efficiency here we combine the benefits of both and propose a low dimensional multiscale modeling of the data at a modest computational cost the idea is to project the classical multidimensional scaling problem into the data spectral domain extracted from its laplace beltrami operator there embedding into a small dimensional euclidean space is accomplished while optimizing for a small number of coefficients we provide a theoretical support and demonstrate that working in the natural eigenspace of the data one could reduce the process complexity while maintaining the model fidelity as examples we efficiently canonize nonrigid shapes by embedding their intrinsic metric into a method often used for matching and classifying almost isometric articulated objects finally we demonstrate the method by exposing the style in which handwritten digits appear in a large collection of images we also visualize clustering of digits by treating images as feature points that we map to a plane

one dimensional k center on uncertain data problems on uncertain data have attracted significant attention due to the imprecise nature of many measurement data in this paper we consider the k center problem on one dimensional uncertain data the input is a set p of weighted uncertain points on a real line and each uncertain point is specified by its probability density function pdf which is a piecewise uniform function i e a histogram the goal is to find a set q of k points on the line to minimize the maximum expected distance from the uncertain points of p to their expected closest points in q we present efficient algorithms for this uncertain k center problem and their running times almost match those for the deterministic k center problem