We consider a range-search variant of the closest-pair problem. Let Gamma be a fixed shape in the plane. We are interested in storing a given set of n points in the plane in some data structure such that for any specified translate of Gamma, the closest pair of points contained in the translate can be reported efficiently. We present results on this problem for two important settings: when Gamma is a polygon (possibly with holes) and when Gamma is a general convex body whose boundary is smooth. When Gamma is a polygon, we present a data structure using O(n) space and O(log n) query time, which is asymptotically optimal. When Gamma is a general convex body with a smooth boundary, we give a near-optimal data structure using O(n log n) space and O(log^2 n) query time. Our results settle some open questions posed by Xue et al. at SoCG 2018.

Given a dataset S of points in R^2, the range closest-pair (RCP) problem aims to preprocess S into a data structure such that when a query range X is specified, the closest-pair in S cap X can be reported efficiently. The RCP problem can be viewed as a range-search version of the classical closest-pair problem, and finds applications in many areas. Due to its non-decomposability, the RCP problem is much more challenging than many traditional range-search problems. This paper revisits the RCP problem, and proposes new data structures for various query types including quadrants, strips, rectangles, and halfplanes. Both worst-case and average-case analyses (in the sense that the data points are drawn uniformly and independently from the unit square) are applied to these new data structures, which result in new bounds for the RCP problem. Some of the new bounds significantly improve the previous results, while the others are entirely new.

In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint/multipoint uncertainty models. Let S=S_R U S_B be a given set of stochastic bichromatic points, and define n = min{|S_R|, |S_B|} and N = max{|S_R|, |S_B|}. We show that the separable-probability (SP) of S can be computed in O(nN^{d-1}) time for d >= 3 and O(min{nN log N, N^2}) time for d=2, while the expected separation-margin (ESM) of S can be computed in O(nN^d) time for d >= 2. In addition, we give an Omega(nN^{d-1}) witness-based lower bound for computing SP, which implies the optimality of our algorithm among all those in this category. Also, a hardness result for computing ESM is given to show the difficulty of further improving our algorithm. As an extension, we generalize the same problems from points to general geometric objects, i.e., polytopes and/or balls, and extend our algorithms to solve the generalized SP and ESM problems in O(nN^d) and O(nN^{d+1}) time, respectively. Finally, we present some applications of our algorithms to stochastic convex-hull related problems.

In a top-k Geometric Intersection Query (top-k GIQ) problem, a set of n weighted, geometric objects in Rd is to be preprocessed into a compact data structure so that for any query geometric object, q, and integer k > 0, the k largest-weight objects intersected by q can be reported efficiently. While the top-k problem has been studied extensively for non-geometric problems (e.g., recommender systems), the geometric version has received little attention. This paper gives a general technique to solve any top-k GIQ problem efficiently. The technique relies only on the availability of an efficient solution for the underlying (non-top-k) GIQ problem, which is often the case. Using this, asymptotically efficient solutions are derived for several top-k GIQ problems, including top-k orthogonal and circular range search, point enclosure search, halfspace range search, etc. Implementations of some of these solutions, using practical data structures, show that they are quite efficient in practice. This paper also does a formal investigation of the hardness of the top-k GIQ problem, which reveals interesting connections between the top-k GIQ problem and the underlying (non-top-k) GIQ problem.

Hadoop, employing the MapReduce programming paradigm, has been widely accepted as the standard framework for analyzing big data in distributed environments. Unfortunately, this rich framework was not truly exploited towards processing large-scale computational geometry operations. This paper introduces CG_Hadoop; a suite of scalable and efficient MapReduce algorithms for various fundamental computational geometry problems, namely, polygon union, skyline, convex hull, farthest pair, and closest pair, which present a set of key components for other geometric algorithms. For each computational geometry operation, CG_Hadoop has two versions, one for the Apache Hadoop system and one for the SpatialHadoop system; a Hadoop-based system that is more suited for spatial operations. These proposed algorithms form a nucleus of a comprehensive MapReduce library of computational geometry operations. Extensive experimental results on a cluster of 25 machines of datasets up to 128GB show that CG_Hadoop achieves up to 29x and 260x better performance than traditional algorithms when using Hadoop and SpatialHadoop systems, respectively.

Let S be a set of n points in Rd where each point has t ≥ 1 real-valued attributes called features. A range-skyline query on S takes as input a query box q ε Rd and returns the skyline of the points of q ∩ S, computed w.r.t. their features (not their coordinates in Rd). Efficient algorithms are given for computing range-skylines and a related hardness result is established.

Mobile robots can be used as mobile hubs to provide communication services on-demand. This capability is especially valuable in disaster response scenarios where there is no communication infrastructure. In such scenarios, mobile hubs can provide a communication infrastructure in a dynamic fashion. In this paper, we study the problem of building a communication bridge between a source s and a destination t with mobile robots. Given a set of robots P and their initial locations, our goal is to find a subset S of robots and their final locations such that the robots in S create a communication bridge between s and t in their final locations. We introduce a new optimization problem for building communication bridges. The objective is to minimize the number of hubs (i.e., |S|), while simultaneously minimizing the robots' motion. The two mobility measures studied in this paper are: (i) maximum travel distance and (ii) total travel distance of the robots. For a geometric version of the problem where the robots must move onto the line segment [s, t], we present polynomial time algorithms which use the minimum number of hubs while remaining within a constant factor of a given motion measure.

BACKGROUND: An algorithm is presented to compute a multiple structure alignment for a set of proteins and to generate a consensus (pseudo) protein which captures common substructures present in the given proteins. The algorithm represents each protein as a sequence of triples of coordinates of the alpha-carbon atoms along the backbone. It then computes iteratively a sequence of transformation matrices (i.e., translations and rotations) to align the proteins in space and generate the consensus. The algorithm is a heuristic in that it computes an approximation to the optimal alignment that minimizes the sum of the pairwise distances between the consensus and the transformed proteins. RESULTS: Experimental results show that the algorithm converges quite rapidly and generates consensus structures that are visually similar to the input proteins. A comparison with other coordinate-based alignment algorithms (MAMMOTH and MATT) shows that the proposed algorithm is competitive in terms of speed and the sizes of the conserved regions discovered in an extensive benchmark dataset derived from the HOMSTRAD and SABmark databases. The algorithm has been implemented in C++ and can be downloaded from the project's web page. Alternatively, the algorithm can be used via a web server which makes it possible to align protein structures by uploading files from local disk or by downloading protein data from the RCSB Protein Data Bank. CONCLUSIONS: An algorithm is presented to compute a multiple structure alignment for a set of proteins, together with their consensus structure. Experimental results show its effectiveness in terms of the quality of the alignment and computational cost.

Determining the influence of an object on other objects in a database, based on proximity, is important in many applications. Abstractly, we wish to pre-process a set, P, of points in d-space so that the points of P that are assigned a new query point q as a Euclidean nearest neighbor can be reported quickly. These are the reverse nearest neighbors of q and are the ones most influenced by q. This generalizes to bichromatic reverse nearest neighbors, in which two sets, clients and servers, are given, where each client is influenced by some server, and of interest are the clients that are assigned a new server q as a nearest neighbor. Both extend to higher orders k > 1, where we seek the points that are assigned q as one of their k nearest neighbors, indicating varying degrees of influence. Each version also has a counterpart where "nearest" is replaced by "farthest", signifying low influence.

Effective diagnosis of Alzheimer's disease (AD) is of primary importance in biomedical research. Recent studies have demonstrated that neuroimaging parameters are sensitive and consistent measures of AD. In addition, genetic and demographic information have also been successfully used for detecting the onset and progression of AD. The research so far has mainly focused on studying one type of data source only. It is expected that the integration of heterogeneous data (neuroimages, demographic, and genetic measures) will improve the prediction accuracy and enhance knowledge discovery from the data, such as the detection of biomarkers. In this paper, we propose to integrate heterogeneous data for AD prediction based on a kernel method. We further extend the kernel framework for selecting features (biomarkers) from heterogeneous data sources. The proposed method is applied to a collection of MRI data from 59 normal healthy controls and 59 AD patients. The MRI data are pre-processed using tensor factorization. In this study, we treat the complementary voxel-based data and region of interest (ROI) data from MRI as two data sources, and attempt to integrate the complementary information by the proposed method. Experimental results show that the integration of multiple data sources leads to a considerable improvement in the prediction accuracy. Results also show that the proposed algorithm identifies biomarkers that play more significant roles than others in AD diagnosis.

Gene expression in a developing embryo occurs in particular cells (spatial patterns) in a time-specific manner (temporal patterns), which leads to the differentiation of cell fates. Images of a Drosophila melanogaster embryo at a given developmental stage, showing a particular gene expression pattern revealed by a gene-specific probe, can be compared for spatial overlaps. The comparison is fundamentally important to formulating and testing gene interaction hypotheses. Expression pattern comparison is most biologically meaningful when images from a similar time point (developmental stage) are compared. In this paper, we present LdaPath, a novel formulation of Linear Discriminant Analysis (LDA) for automatic developmental stage range classification. It employs multivariate linear regression with the L(1)-norm penalty controlled by a regularization parameter for feature extraction and visualization. LdaPath computes an entire solution path for all values of regularization parameter with essentially the same computational cost as fitting one LDA model. Thus, it facilitates efficient model selection. It is based on the equivalence relationship between LDA and the least squares method for multi-class classifications. This equivalence relationship is established under a mild condition, which we show empirically to hold for many high-dimensional datasets, such as expression pattern images. Our experiments on a collection of 2705 expression pattern images show the effectiveness of the proposed algorithm. Results also show that the LDA model resulting from LdaPath is sparse, and irrelevant features may be removed. Thus, LdaPath provides a general framework for simultaneous feature selection and feature extraction.

Classical Linear Discriminant Analysis (LDA) is not applicable for small sample size problems due to the singularity of the scatter matrices involved. Regularized LDA (RLDA) provides a simple strategy to overcome the singularity problem by applying a regularization term, which is commonly estimated via cross-validation from a set of candidates. However, cross-validation may be computationally prohibitive when the candidate set is large. An efficient algorithm for RLDA is presented that computes the optimal transformation of RLDA for a large set of parameter candidates, with approximately the same cost as running RLDA a small number of times. Thus it facilitates efficient model selection for RLDA. An intrinsic relationship between RLDA and Uncorrelated LDA (ULDA), which was recently proposed for dimension

Dimension reduction is critical in many areas of data mining and machine learning. In this paper, a Covariance-preserving Projection Method (CPM for short) is proposed for dimension reduction. CPM maximizes the class discrimination and also preserves approximately the class covariance. The optimization involved in CPM can be formulated as low rank approximations of a collection of matrices, which can be solved iteratively. Our theoretical and empirical analysis reveals the relationship between CPM and Linear Discriminant Analysis (LDA), Sliced Average Variance Estimator (SAVE), and Heteroscedastic Discriminant Analysis (HDA). This gives us new insights into the nature of these different algorithms. We use both synthetic and real-world datasets to evaluate the effectiveness of the proposed algorithm.

High-dimensional data appear in many applications of data mining, machine learning, and bioinformatics. Feature reduction is commonly applied as a preprocessing step to overcome the curse of dimensionality. Uncorrelated Linear Discriminant Analysis (ULDA) was recently proposed for feature reduction. The extracted features via ULDA were shown to be statistically uncorrelated, which is desirable for many applications. In this paper, an algorithm called ULDA/QR is proposed to simplify the previous implementation of ULDA. Then, the ULDA/GSVD algorithm is proposed, based on a novel optimization criterion, to address the singularity problem which occurs in undersampled problems, where the data dimension is larger than the sample size. The criterion used is the regularized version of the one in ULDA/QR. Surprisingly, our theoretical result shows that the solution to ULDA/GSVD is independent of the value of the regularization parameter. Experimental results on various types of data sets are reported to show the effectiveness of the proposed algorithm and to compare it with other commonly used feature reduction algorithms.

Monitoring traffic intersections in real-time and predicting possible collisions is an important first step towards building an early collision warning system. We present the general vision methods used in a system addressing this problem and describe the practical adaptations necessary to achieve real-time performance. A novel method for three-dimensional vehicle size estimation is presented. We also describe a method for target localization in real-world coordinates which allows for sequential incorporation of measurements from multiple cameras into a single target's state vector. Additionally, a fast implementation of a false-positive reduction method for the foreground pixel masks is developed. Finally, a low-overhead collision prediction algorithm using the time-as-axis paradigm is presented. The proposed system was able to perform in real-time on videos of quarter-VGA (320/spl times/240) resolution. The errors in target position and dimension estimates in a test video sequence are quantified.

Monitoring traffic intersections in real time and predicting possible collisions is an important first step towards building an early collision-warning system. We present a vision-based system addressing this problem and describe the practical adaptations necessary to achieve real-time performance. Innovative low-overhead collision-prediction algorithms (such as the one using the time-as-axis paradigm) are presented. The proposed system was able to perform successfully in real time on videos of quarter-video graphics array (VGA) (320 /spl times/ 240) resolution under various weather conditions. The errors in target position and dimension estimates in a test video sequence are quantified and several experimental results are presented.

Dimension reduction is a critical data preprocessing step for many database and data mining applications, such as efficient storage and retrieval of high-dimensional data. In the literature, a well-known dimension reduction algorithm is linear discriminant analysis (LDA). The common aspect of previously proposed LDA-based algorithms is the use of singular value decomposition (SVD). Due to the difficulty of designing an incremental solution for the eigenvalue problem on the product of scatter matrices in LDA, there has been little work on designing incremental LDA algorithms that can efficiently incorporate new data items as they become available. In this paper, we propose an LDA-based incremental dimension reduction algorithm, called IDR/QR, which applies QR decomposition rather than SVD. Unlike other LDA-based algorithms, this algorithm does not require the whole data matrix in main memory. This is desirable for large data sets. More importantly, with the insertion of new data items, the IDR/QR algorithm can constrain the computational cost by applying efficient QR-updating techniques. Finally, we evaluate the effectiveness of the IDR/QR algorithm in terms of classification error rate on the reduced dimensional space. Our experiments on several real-world data sets reveal that the classification error rate achieved by the IDR/QR algorithm is very close to the best possible one achieved by other LDA-based algorithms. However, the IDR/QR algorithm has much less computational cost, especially when new data items are inserted dynamically.

Feature extraction is important in many applications, such as text and image retrieval, because of high dimensionality. Uncorrelated Linear Discriminant Analysis (ULDA) was recently proposed for feature extraction. The extracted features via ULDA were shown to be statistically uncorrelated, which is desirable for many applications. In this paper, we will first propose the ULDA/QR algorithm to simplify the previous implementation of ULDA. Then we propose the ULDA/GSVD algorithm, based on a novel optimization criterion, to address the singularity problem. It is applicable for undersampled problem, where the data dimension is much larger than the data size, such as text and image retrieval. The novel criterion used in ULDA/GSVD is the perturbed version of the one from ULDA/QR, while surprisingly, the solution to ULDA/GSVD is shown to be independent of the amount of perturbation applied. We did extensive experiments on text and face image data to show the effectiveness of ULDA/GSVD and compare with other popular feature extraction algorithms.

Recent years have witnessed a dramatic increase in the quantity of image data collected, due to advances in fields such as medical imaging, reconnaissance, surveillance, astronomy, multimedia etc. With this increase has come the need to be able to store, transmit, and query large volumes of image data efficiently. A common operation on image databases is the retrieval of all images that are similar to a query image. For this, the images in the database are often represented as vectors in a high-dimensional space and a query is answered by retrieving all image vectors that are proximal to the query image in this space, under a suitable similarity metric. To overcome problems associated with high dimensionality, such as high storage and retrieval times, a dimension reduction step is usually applied to the vectors to concentrate relevant information in a small number of dimensions. Principal Component Analysis (PCA) is a well-known dimension reduction scheme. However, since it works with vectorized representations of images, PCA does not take into account the spatial locality of pixels in images. In this paper, a new dimension reduction scheme, called Generalized Principal Component Analysis (GPCA), is presented. This scheme works directly with images in their native state, as two-dimensional matrices, by projecting the images to a vector space that is the tensor product of two lower-dimensional vector spaces. Experiments on databases of face images show that, for the same amount of storage, GPCA is superior to PCA in terms of quality of the compressed images, query precision, and computational cost.

Dimension reduction is critical for many database and data mining applications, such as efficient storage and retrieval of high-dimensional data. In the literature, a well-known dimension reduction scheme is Linear Discriminant Analysis (LDA). The common aspect of previously proposed LDA based algorithms is the use of Singular Value Decomposition (SVD). Due to the difficulty of designing an incremental solution for the eigenvalue problem on the product of scatter matrices in LDA, there is little work on designing incremental LDA algorithms. In this paper, we propose an LDA based incremental dimension reduction algorithm, called IDR/QR, which applies QR Decomposition rather than SVD. Unlike other LDA based algorithms, this algorithm does not require the whole data matrix in main memory. This is desirable for large data sets. More importantly, with the insertion of new data items, the IDR/QR algorithm can constrain the computational cost by applying efficient QR-updating techniques. Finally, we evaluate the effectiveness of the IDR/QR algorithm in terms of classification accuracy on the reduced dimensional space. Our experiments on several real-world data sets reveal that the accuracy achieved by the IDR/QR algorithm is very close to the best possible accuracy achieved by other LDA based algorithms. However, the IDR/QR algorithm has much less computational cost, especially when new data items are dynamically inserted.

Linear Discriminant Analysis (LDA) is a well-known method for fea- ture extraction and dimension reduction. It has been used widely in many applications such as face recognition. Recently, a novel LDA algo- rithm based on QR Decomposition, namely LDA/QR, has been proposed, which is competitive in terms of classification accuracy with other LDA algorithms, but it has much lower costs in time and space. However, LDA/QR is based on linear projection, which may not be suitable for data with nonlinear structure. This paper first proposes an algorithm called KDA/QR, which extends the LDA/QR algorithm to deal with nonlin- ear data by using the kernel operator. Then an efficient approximation of KDA/QR called AKDA/QR is proposed. Experiments on face image data show that the classification accuracy of both KDA/QR and AKDA/QR are competitive with Generalized Discriminant Analysis (GDA), a gen- eral kernel discriminant analysis algorithm, while AKDA/QR has much lower time and space costs. 1 Introduction Linear Discriminant Analysis [3] is a wellknown method for dimension reduction. It has been used widely in many applications such as face recognition [2]. Classical LDA aims to find optimal transformation by minimizing the within-class distance and maximizing the between-class distance simultaneously, thus achieving maximum discrimination. The optimal transformation can be readily computed by computing the eigen-decomposition on the scatter matrices. Although LDA works well for linear problems, it may be less effective when severe non- linearity is involved. To deal with such a limitation, nonlinear extensions through kernel functions have been proposed. The main idea of kernel-based methods is to map the input data to a feature space through a nonlinear mapping, where the inner products in the feature space can be computed by a kernel function without knowing the nonlinear mapping explic- itly [9]. Kernel Principal Component Analysis (KPCA) [10], Kernel Fisher Discriminant Analysis (KFDA) [7] and Generalized Discriminant Analysis (GDA) [1] are, respectively, kernel-based nonlinear extensions of the well known PCA, FDA and LDA methods. To our knowledge, there are few efficient algorithms for general kernel based discriminant algorithms -- most known algorithms effectively scale as O(n3) where n is the sample size. In [6, 8], S. Mika et al. made a first attempt to speed up KFDA through a greedy approximation technique. However the algorithm was developed to handle the binary clas- sification problem. For multi-class problem, the authors suggested the one against the rest scheme by considering all two-class problems. Recently, an efficient variant of LDA, namely LDA/QR, was proposed in [11, 12]. The essence of LDA/QR is the utilization of QR-decomposition on a small size matrix. The time complexity of LDA/QR is linear in the size of the training data, as well as the number of dimensions of the data. Moreover, experiments in [11, 12] show that the classification accuracy of LDA/QR is competitive with other LDA algorithms. In this paper, we first propose an algorithm, namely KDA/QR1, which is a nonlinear exten- sion of LDA/QR. Since KDA/QR involves the whole kernel matrix, which is not scalable for large datasets, we also propose an approximation of KDA/QR, namely AKDA/QR. A distinct property of AKDA/QR is that it scales as O(ndc), where n is the size of the data, d is the dimension of the data, and c is the number of classes. We apply the proposed algorithms on face image datasets and compare them with LDA/QR, and Generalized Discriminant Analysis (GDA) [1], a general method for kernel discrim- inant analysis. Experiments show that: (1) AKDA/QR is competitive with KDA/QR and GDA in classification; (2) both KDA/QR and AKDA/QR outperform LDA/QR in classifi- cation; and (3) AKDA/QR has much lower costs in time and space than GDA.

Linear Discriminant Analysis (LDA) is a well-known scheme for feature extraction and dimension reduction. It has been used widely in many ap- plications involving high-dimensional data, such as face recognition and image retrieval. An intrinsic limitation of classical LDA is the so-called singularity problem, that is, it fails when all scatter matrices are singu- lar. A well-known approach to deal with the singularity problem is to apply an intermediate dimension reduction stage using Principal Com- ponent Analysis (PCA) before LDA. The algorithm, called PCA+LDA, is used widely in face recognition. However, PCA+LDA has high costs in time and space, due to the need for an eigen-decomposition involving the scatter matrices. In this paper, we propose a novel LDA algorithm, namely 2DLDA, which stands for 2-Dimensional Linear Discriminant Analysis. 2DLDA over- comes the singularity problem implicitly, while achieving efficiency. The key difference between 2DLDA and classical LDA lies in the model for data representation. Classical LDA works with vectorized representa- tions of data, while the 2DLDA algorithm works with data in matrix representation. To further reduce the dimension by 2DLDA, the combi- nation of 2DLDA and classical LDA, namely 2DLDA+LDA, is studied, where LDA is preceded by 2DLDA. The proposed algorithms are ap- plied on face recognition and compared with PCA+LDA. Experiments show that 2DLDA and 2DLDA+LDA achieve competitive recognition accuracy, while being much more efficient.

An optimization criterion is presented for discriminant analysis. The criterion extends the optimization criteria of the classical Linear Discriminant Analysis (LDA) through the use of the pseudoinverse when the scatter matrices are singular. It is applicable regardless of the relative sizes of the data dimension and sample size, overcoming a limitation of classical LDA. The optimization problem can be solved analytically by applying the Generalized Singular Value Decomposition (GSVD) technique. The pseudoinverse has been suggested and used for undersampled problems in the past, where the data dimension exceeds the number of data points. The criterion proposed in this paper provides a theoretical justification for this procedure. An approximation algorithm for the GSVD-based approach is also presented. It reduces the computational complexity by finding subclusters of each cluster and uses their centroids to capture the structure of each cluster. This reduced problem yields much smaller matrices to which the GSVD can be applied efficiently. Experiments on text data, with up to 7,000 dimensions, show that the approximation algorithm produces results that are close to those produced by the exact algorithm.

The classification of tissue samples based on gene expression data is an important problem in medical diagnosis of diseases such as cancer. In gene expression data, the number of genes is usually very high (in the thousands) compared to the number of data samples (in the tens or low hundreds); that is, the data dimension is large compared to the number of data points (such data is said to be undersampled). To cope with performance and accuracy problems associated with high dimensionality, it is commonplace to apply a preprocessing step that transforms the data to a space of significantly lower dimension with limited loss of the information present in the original data. Linear Discriminant Analysis (LDA) is a well-known technique for dimension reduction and feature extraction, but it is not applicable for undersampled data due to singularity problems associated with the matrices in the underlying representation. This paper presents a dimension reduction and feature extraction scheme, called Uncorrelated Linear Discriminant Analysis (ULDA), for undersampled problems and illustrates its utility on gene expression data. ULDA employs the Generalized Singular Value Decomposition method to handle undersampled data and the features that it produces in the transformed space are uncorrelated, which makes it attractive for gene expression data. The properties of ULDA are established rigorously and extensive experimental results on gene expression data are presented to illustrate its effectiveness in classifying tissue samples. These results provide a comparative study of various state-of-the-art classification methods on well-known gene expression data sets.

A new optimization criterion for discriminant analysis is presented. The new criterion extends the optimization criteria of the classical linear discriminant analysis (LDA) by introducing the pseudo-inverse when the scatter matrices are singular. It is applicable regardless of the relative sizes of the data dimension and sample size, overcoming a limitation of the classical LDA. Recently, a new algorithm called LDA/GSVD for structure-preserving dimension reduction has been introduced, which extends the classical LDA to very high-dimensional undersampled problems by using the generalized singular value decomposition (GSVD). The solution from the LDA/GSVD algorithm is a special case of the solution for our generalized criterion, which is also based on GSVD. We also present an approximate solution for our GSVD-based solution, which reduces computational complexity by finding subclusters of each cluster, and using their centroids to capture the structure of each cluster. This reduced problem yields much smaller matrices of which the GSVD can be applied efficiently. Experiments on text data, with up to 7000 dimensions, show that the approximation algorithm produces results that are close to those produced by the exact algorithm.

Determining structural similarities between proteins is animportant problem since it can help identify functional and evolutionary relationships. In this paper, an algorithm is proposed to align two protein structures. Given the protein backbones, the algorithm finds a rigid motion of one backbone onto the other such that large substructures are matched. The algorithm uses a representation of the backbone that is independent of their relative orientations in space and applies dynamic programming to this representation to compute an initial alignment, which is then refined iteratively. Experiments indicate that the algorithm is competitive with two well-known algorithms, namely DALI [12] and LOCK [19].

In Layered Manufacturing, the choice of the build direction for the model inuences several design criteria, including the number of layers, the volume and contact-area of the support structures, and the surface nish. These, in turn, impact the throughput and cost of the process. In this paper, ecient geometric algorithms are given to reconcile two or more of these criteria simultaneously, under three formulations of multi-criteria optimization: Finding a build direction which (i) optimizes the criteria sequentially, (ii) optimizes their weighted sum, or (iii) allows the criteria to meet designer-prescribed thresholds. The algorithms that involve \\support volume" or \\contact area" apply only to convex models, while the algorithms that involve \\surface nish" and \\number of layers" apply to any polyhedral model. Some of the latter algorithms have also been implemented and tested on real-world models obtained from industry. The geometric techniques used in the paper include construction and searching of certain arrangements on the unit-sphere, three-dimensional convex hulls, Voronoi diagrams, point location, and hierarchical representations. Additionally, solutions are also provided, for the constrained versions of two geometric problems, namely polyhedron width and largest empty disk on the unit-sphere. 1

Net extraction is crucial in VLSI design verification. Current algorithms for net extraction do not exploit the fact that the number, c, of different orientations of the line segments or polygons in a practical VLSI mask design is small relative to the number, n, of segments or polygon edges. Instead they rely on computing all intersections in the input and hence take time that is at least proportional to the number of intersections. In this paper we develop and implement a practical algorithm for net extraction that runs in O(cn log n) time and O(n) space, which is optimal for fixed c. The algorithm uses only integer operations and is, as a result, numerically stable. Experiments indicate that the algorithm will outperform existing algorithms on practical VLSI designs. We expect that the techniques presented will be useful in other VLSI/CAD problems that operate with restricted orientation geometries.

Generalized intersection searching problems are a class of geometric query-retrieval problems where the questions of interest concern the intersection of a query object with aggregates of geometric objects (rather than with individual objects.) This class contains, as a special case, the well-studied class of standard intersection searching problems and is rich in applications. Unfortunately, the solutions known for the standard problems do not yield efficient solutions to the generalized problems. Recently, efficient solutions have been given for generalized problems where the input and query objects are iso-oriented (i.e., axes-parallel) or where the aggregates satisfy additional properties (e.g., connectedness). In this paper, efficient algorithms are given for several generalized problems involving non-iso-oriented objects. These problems include: generalized halfspace range searching, segment intersection searching, triangle stabbing, and triangle range searching. The techniques used include: computing suitable sparse representations of the input, persistent data structures, and filtering search.

Net extraction is crucial in VLSI design verification. Current algorithms for net extraction do not exploit the fact that the number, c, of different orientations of the line segments or polygon edges in a practical VLSI mask design is small relative to the number, n, of segments or edges. Instead, they rely on computing all intersections in the input and hence take time that is at least proportional to the number of intersections. In this paper we develop a simple and practical algorithm for net extraction that runs in O(cn log n) time and O(n) space, which is optimal for fixed c. Experiments indicate that the algorithm will generally outperform existing algorithms on practical VLSI designs. We expect that the techniques presented will be useful in other VLSI CAD problems that operate with restricted orientation geometries.

A point location scheme is presented for a dynamic planar subdivision whose underlying graph is only required to be connected. The operations supported include: reporting the name of the region containing a query point, inserting/deleting an edge, and inserting/deleting/moving a degree-2 vertex. The scheme uses $O(n)$ space, has a worst-case query time of $O(\log ^2 n)$, and a worst-case update time of $O(\log n)$, where n is the number of vertices currently in the subdivision. Insertion (respectively, deletion) of an arbitrary k-edge chain inside a region can be performed in $O(k\log (n + k))$ (respectively, $O(k\log n)$) time in worst-case. The scheme outperforms the solutions given in works by Fries, Mehlhorn, and Naeher and by Overmars and also handles more general subdivisions than the schemes given in works by Preparata and Tamassia. The result is based on a new solution to a dynamic visibility problem for a set of line segments in the plane that are nonintersecting, except possibly at endpoints. The scheme is then extended to speed up the insertion/deletion of a k-edge monotone chain to $O(\log ^2 n\log \log n + k)$ time (or $O(\log n\log \log n + k)$ time for an alternative model of input), but at the expense of increasing the other time bounds slightly. Additional results include a generalization to subdivisions consisting of algebraic segments of bounded degree and a persistent scheme that allows point location queries in the past and updates in the present.

A point location scheme is presented for an n-vertex dynamic planar subdivision whose underlying graph is only required to be connected. The scheme uses O(n) space and yields an O(log/sup 2/n) query time and an O(log n) update time. Insertion (respectively, deletion) of an arbitrary k-edge chain inside a region can be performed in O(k log(n+k)) (respectively, O(k log n)) time. The scheme is then extended to speed up the insertion/deletion of a k-edge monotone chain to O(log/sup 2/n log log n+k) time (or O(log n log log n+k) time for an alternative model of input), but at the expense of increasing the other time bounds slightly. All bounds are worst case. Additional results include a generalization to planar subdivisions consisting of algebraic segments of bounded degree and a persistent scheme for planar point location.>

The problem of routing messages along near-shortest paths in a distributed network without using complete routing tables is considered. It is assumed that the nodes of the network can be assigned suitable short names at the time the network is established. Two space-efficient near- shortest path routing schemes are given for any class of networks whose members can be decomposed recursively by a separator of size at most a constant c, where $c \geqslant 2$. For an n-node network, the first scheme uses a total of $O(cn \log n)$ items of routing information, each $O(\log n)$ bits long, and $O(\log n)$-bit names, generated from a separator-based decomposition of the network, to achieve routings that are at most three times longer than shortest routings in worst case.l The second scheme augments the node names with $O(c \log c \log n)$ additional bits and uses this to reduce the bound on the routings to $(2 / \alpha) + 1$, where $\alpha, 1 < \alpha \leq 2$, is the root of the equation $\alpha ^{\lceil (c+1) / 2\rceil } - \alpha - 2 = 0$. For both schemes, the node names and the routing information can be determined efficiently.

The problem of routing messages along near-shortest paths in a distributed network without using complete routing tables is considered. It is assumed that the nodes of the network can be assigned suitable short names at the time the network is established. Two space-efficient near-shortest-path routing schemes are given for the class of planar networks. Both schemes use the separator property of planar networks in assigning the node names and performing the routings. For an n-node network, the first scheme uses $O(\log n)$-bit names and a total of $O(n^{{4 / 3}} )$ items of routing information, each $O(\log n)$ bits long, to generate routings that are only three times longer than corresponding shortest routings in worst cases. For any constant $\epsilon ,0 < \epsilon < 1/3$, the second scheme achieves the better space bound of $O(n^{1 + \epsilon } )$ items, each $O(({1 / \epsilon })\log n)$ bits long, but at the expense of $O(({1 / \epsilon })\log n)$-bit node names and a worst-case bound of 7 on the routings.

The problem of designing space- and communication-efficient routing schemes for networks that experience faults is addressed. For any outerplanar network containing t faults, a succinct routing scheme is presented that uses O( alpha tn) space and communication to generate routings that are less than (( alpha +1)/( alpha -1))/sup t/ times longer than optimal, where alpha >1 is an odd-valued integer parameter. Thus, the routings can be tuned as desired, using a suitable amount of information. Efficient sequential and distributed algorithms are presented for setting up the routing schemes.>

Message routing strategies are given for networks with certain separator properties. These strategies use considerably less space than complete routing tables, keep node names to O(log n) bits, and still route along near-shortest paths. For any network with separators of size at most a small constant c, a total of O(n log n) items of routing information is stored, and any message is routed along a path of length at most (2/α) + 1 times the length of an optimal path, where α ≫ 1 is the positive root of the equation α⌈(c+1)/2⌉ - α - 2 = 0. For planar networks, O(n1+ε) items are stored, for any constant ε, 0 ≪ ε ≪ 1/3, and the length of any message path is at most 7 times that of an optimal path.

Article Free Access Share on Optimal message routing without complete routing tables Authors: Greg N Frederickson Department of Computer Science, Purdue University, West Lafayette, Indiana Department of Computer Science, Purdue University, West Lafayette, IndianaView Profile , Ravi Janardan Department of Computer Science, Purdue University, West Lafayette, Indiana Department of Computer Science, Purdue University, West Lafayette, IndianaView Profile Authors Info & Claims PODC '86: Proceedings of the fifth annual ACM symposium on Principles of distributed computingNovember 1986 Pages 88–97https://doi.org/10.1145/10590.10598Published:01 November 1986Publication History 9citation207DownloadsMetricsTotal Citations9Total Downloads207Last 12 Months7Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF