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### Separating Geometric Data with Minimum Cost: Two Disjoint Convex Hulls [article]

2021 arXiv   pre-print
Furthermore, it has been illustrated that the new problem known as "Two Disjoint Convex Hulls" can be solved in polynomial time due to some combinatorial aspects and geometric properties.  ...  In this study, a geometric version of an NP-hard problem ("Almost 2-SAT" problem) is introduced which has potential applications in clustering, separation axis, binary sensor networks, shape separation  ...  It chooses the pair with the minimum cost and returns the points that should make the separating axis.  ...

### A general soft method for learning SVM classifiers with -norm penalty

Qing Tao, Gao-Wei Wu, Jue Wang
2008 Pattern Recognition
Intrinsically, the resulting soft algorithms are to find -optimal nearest points between two soft convex hulls.  ...  Based on the geometric interpretation of support vector machines (SVMs), this paper presents a general technique that allows almost all the existing L 2 -norm penalty based geometric algorithms, including  ...  Hence, in this paper, it is natural to assume that the two soft convex hulls are separable with a choice of .  ...

### Shooting Permanent Rays among Disjoint Polygons in the Plane

Mashhood Ishaque, Bettina Speckmann, Csaba D. Tóth
2012 SIAM journal on computing (Print)
obstacles with a total of n vertices in the plane, a permutation of the reflex vertices, and a half-line at each reflex vertex that partitions the reflex angle into two convex angles, then the convex partitioning  ...  We present a data structure for ray shooting and insertion in the free space between disjoint polygonal obstacles with a total of n vertices in the plane, where each ray starts at the boundary of some  ...  For a set S of points in the plane, the convex hull is the minimum set that contains S and is convex (i.e., contains the line segment between any two of its points).  ...

### Shooting permanent rays among disjoint polygons in the plane

Mashhood Ishaque, Bettina Speckmann, Csaba D. Tóth
2009 Proceedings of the 25th annual symposium on Computational geometry - SCG '09
obstacles with a total of n vertices in the plane, a permutation of the reflex vertices, and a half-line at each reflex vertex that partitions the reflex angle into two convex angles, then the convex partitioning  ...  We present a data structure for ray shooting and insertion in the free space between disjoint polygonal obstacles with a total of n vertices in the plane, where each ray starts at the boundary of some  ...  For a set S of points in the plane, the convex hull is the minimum set that contains S and is convex (i.e., contains the line segment between any two of its points).  ...

### Linear Separability in Spatial Databases [article]

Gilberto Gutiérrez, Pablo Pérez-Lantero, Claudio Torres
2016 arXiv   pre-print
Given two point sets R and B in the plane, with cardinalities m and n, respectively, and each set stored in a separate R-tree, we present an algorithm to decide whether R and B are linearly separable.  ...  We also present an algorithm to compute the convex hull of n planar points given in an R-tree, running in O(n n) time in the worst case.  ...  Among the geometric problems in spatial databases, this later work is well related to the results of this paper since linear the separability of two point sets can be decided by computing the convex hull  ...

### Cluster Hull: A Technique for Summarizing Spatial Data Streams

J. Hershberger, N. Shrivastava, Subhash Suri
2006 22nd International Conference on Data Engineering (ICDE'06)
The convex hulls need not be disjoint, but together these hulls are limited to m vertices, which is our memory budget.  ...  The algorithm attempts to maintain a partition of the point stream with minimum total cost, using its fixed O(k) memory budget.  ...

### Efficient geometric algorithms for parsing in two dimensions

P. Liang, M. Narasimhan, M. Shilman, P. Viola
2005 Eighth International Conference on Document Analysis and Recognition (ICDAR'05)
Each data structure embodies a different geometrical constraint on the set of possible valid parses.  ...  Assuming a clean design for the parsing software, the same parsing framework can be tested with various geometric constraints to determine the most effective combination.  ...  Essentially the recurrence states that the score for A is computed by finding a low cost decomposition of the terminals into two disjoint sets.  ...

### Data Gathering by Mobile Mules in a Spatially Separated Wireless Sensor Network

Fang-Jing Wu, Chi-Fu Huang, Yu-Chee Tseng
2009 2009 Tenth International Conference on Mobile Data Management: Systems, Services and Middleware
In this paper, we consider the possibility that a WSN may be spatially separated into multiple subnetworks.  ...  Data gathering, which is a fundamental mission of WSN, thus may rely on a mobile mule ("mule" for short) to conduct data gathering by visiting each subnetwork.  ...  In addition, the convex hull never intersects with itself. These observations motivate us to conduct path construction and path improvement based on convex hulls.  ...

### Adaptive sampling for geometric problems over data streams

John Hershberger, Subhash Suri
2004 Proceedings of the twenty-third ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems - PODS '04
One can also pose queries over multiple streams: track the minimum distance between the convex hulls of two data streams; or report when datasets A and B are no longer linearly separable.  ...  With our sample convex hull, all the queries mentioned above can be answered in either O(log r) or O(r) time.  ...  Linear Separation: Given approximate convex hulls for two point streams, A and B, we can track the minimum distance between them, or determine when they are no longer linearly separable.  ...

### Summarizing Spatial Data Streams Using ClusterHulls

John Hershberger, Nisheeth Shrivastava, Subhash Suri
2006 2006 Proceedings of the Eighth Workshop on Algorithm Engineering and Experiments (ALENEX)
We are not aware of other schemes specifically designed for capturing shape information in geometric data streams, and so we compare Cluster-Hull with some of the best general-purpose clustering schemes  ...  We propose a novel scheme, called ClusterHull, which represents the shape of the stream as a dynamic collection of convex hulls, with a total of at most m vertices, where m is the size of the memory.  ...  The first 2k points that arrive in a data stream are paired up into k two-point convex hulls, each with cost zero, and the pairing will be arbitrary.  ...

### Summarizing spatial data streams using ClusterHulls

John Hershberger, Nisheeth Shrivastava, Subhash Suri
2009 ACM Journal of Experimental Algorithmics
We are not aware of other schemes specifically designed for capturing shape information in geometric data streams, and so we compare Cluster-Hull with some of the best general-purpose clustering schemes  ...  We propose a novel scheme, called ClusterHull, which represents the shape of the stream as a dynamic collection of convex hulls, with a total of at most m vertices, where m is the size of the memory.  ...  The first 2k points that arrive in a data stream are paired up into k two-point convex hulls, each with cost zero, and the pairing will be arbitrary.  ...

### Kinetic Data Structures [chapter]

2006 Geometric Data Structures for Computer Graphics
A number of other authors have dealt with geometric problems arising from motion, such as collision detection or minimum separation determination [GJS96, ST95, ST96] .  ...  For two moving convex polygons in the plane, a kinetic algorithm where the number of events is a function of the relative separation of the two polygons is given in [EGSZ99] .  ...

### Kinetic Data Structures [chapter]

Leonidas Guibas
2004 Handbook of Data Structures and Applications
A number of other authors have dealt with geometric problems arising from motion, such as collision detection or minimum separation determination [GJS96, ST95, ST96] .  ...  For two moving convex polygons in the plane, a kinetic algorithm where the number of events is a function of the relative separation of the two polygons is given in [EGSZ99] .  ...

### Convex and concave hulls for classification with support vector machine

Asdrúbal López Chau, Xiaoou Li, Wen Yu
2013 Neurocomputing
The proposed convex-concave hull SVM classifier has distinctive advantages on dealing with large data sets with higher accuracy.  ...  The training of a support vector machine (SVM) has the time complexity of O(n 3 ) with data number n . Normal SVM algorithms are not suitable for classification of large data sets.  ...  Convex and Concave Hull Convex Hull The convex hull (CH) of a set of points S is the minimum convex set that contains S.  ...

### Adaptive sampling for geometric problems over data streams

John Hershberger, Subhash Suri
2008 Computational geometry
One can also pose queries over multiple streams: for instance, track the minimum distance between the convex hulls of two data streams, report when datasets A and B are no longer linearly separable, or  ...  Geometric coordinates are an integral part of many data streams.  ...  the convex hulls of two data streams; report when data streams A and B are no longer linearly separable; track the overlap between the spatial extents of streams A and B; report when points of data stream  ...
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