Practical methods for shape fitting and kinetic data structures using core sets

Yu, Hai; Agarwal, Pankaj K.; Poreddy, Raghunath; Varadarajan, Kasturi

doi:10.1145/997817.997858

Cited by 20 publications

(23 citation statements)

References 18 publications

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“…One reason that the running time of our implementation does not seem dependent exponentially on n in practice is because of the fact that the core-set size seems to be dependent on k and the dimension but independent of n for most real life inputs. This has also been observed in other implementations based on core-sets [35,53]. There are some optimizations that we implemented in the algorithm.…”

supporting

confidence: 73%

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ALMOST OPTIMAL SOLUTIONS TO k-CLUSTERING PROBLEMS

Kumar

2010

Int. J. Comput. Geom. Appl.

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supporting

confidence: 73%

“…Previous Work : We do not know of any attempts to implement a solution to the k-clustering problem (except when k = 1) [35,53]. Most previous work related to k-clustering has been focused on variants of the k-center problem and the k-line-center problem [2,3,10,20,28,35].…”

Section: Definition (K-clustering)mentioning

confidence: 99%

ALMOST OPTIMAL SOLUTIONS TO k-CLUSTERING PROBLEMS

Kumar

2010

Int. J. Comput. Geom. Appl.

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“…However, special cases exist where |C Q | = |Q|, e.g., all points in Q are vertices of a convex polygon. To handle such special instances, we propose to obtain an approximate convex hull of Q using Dudley's approximation [35]. Dudley's construction generates an approximate convex hull of Q (denote it as AC Q ) with O(1/ǫ (d−1)/2 ) vertices with maximum Hausdorff distance of ǫ to the convex hull of Q.…”

Section: Handling Disk-resident Query Groupsmentioning

confidence: 99%

“…Dudley's approximation was originally proposed for main memory data sets [35]. For our purpose, it needs to work with external data sets.…”

Section: Handling Disk-resident Query Groupsmentioning

confidence: 99%

Group Enclosing Queries

Yao

Kumar

2011

IEEE Trans. Knowl. Data Eng.

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Abstract-Given a set of points P and a query set Q, a group enclosing query (GEQ) fetches the point p * ∈ P such that the maximum distance of p * to all points in Q is minimized. This problem is equivalent to the Min-Max case (minimizing the maximum distance) of aggregate nearest neighbor queries for spatial databases [27]. This work first designs a new exact solution by exploring new geometric insights, such as the minimum enclosing ball, the convex hull and the furthest voronoi diagram of the query group. To further reduce the query cost, especially when the dimensionality increases, we turn to approximation algorithms. Our main approximation algorithm has a worst case √ 2-approximation ratio if one can find the exact nearest neighbor of a point. In practice, its approximation ratio never exceeds 1.05 for a large number of data sets up to six dimension. We also discuss how to extend it to higher dimensions (up to 74 in our experiment) and show that it still maintains a very good approximation quality (still close to 1) and low query cost. In fixed dimensions, we extend the √ 2-approximation algorithm to get a (1 + ǫ)-approximate solution for the GEQ problem. Both approximation algorithms have O(log N + M ) query cost in any fixed dimension, where N and M are the sizes of the data set P and query group Q. Extensive experiments on both synthetic and real data sets, up to 10 million points and 74 dimensions, confirm the efficiency, effectiveness and scalability of the proposed algorithms, especially their significant improvement over the state-of-the-art method.

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“…However, in some rare cases |CQ| could be close to |Q|. When it does happen, we can obtain an approximate convex hull of Q efficiently using Dudley's approximation based on the coreset idea [24].…”

Section: When Q Is Largementioning

confidence: 99%

Flexible aggregate similarity search

et al. 2011

Proceedings of the 2011 ACM SIGMOD International Conference on Management of Data

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Aggregate similarity search, a.k.a. aggregate nearest neighbor (Ann) query, finds many useful applications in spatial and multimedia databases. Given a group Q of M query objects, it retrieves the most (or top-k) similar object to Q from a database P , where the similarity is an aggregation (e.g., sum, max) of the distances between the retrieved object p and all the objects in Q. In this paper, we propose an added flexibility to the query definition, where the similarity is an aggregation over the distances between p and any subset of φM objects in Q for some support 0 < φ ≤ 1. We call this new definition flexible aggregate similarity (Fann) search, which generalizes the Ann problem. Next, we present algorithms for answering Fann queries exactly and approximately. Our approximation algorithms are especially appealing, which are simple, highly efficient, and work well in both low and high dimensions. They also return nearoptimal answers with guaranteed constant-factor approximations in any dimensions. Extensive experiments on large real and synthetic datasets from 2 to 74 dimensions have demonstrated their superior efficiency and high quality.

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Practical methods for shape fitting and kinetic data structures using core sets

Cited by 20 publications

References 18 publications

ALMOST OPTIMAL SOLUTIONS TO k-CLUSTERING PROBLEMS

ALMOST OPTIMAL SOLUTIONS TO k-CLUSTERING PROBLEMS

Group Enclosing Queries

Flexible aggregate similarity search

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