In many applications it is desirable to cluster high dimensional data along various subspaces, which we refer to as projective clustering. We propose a new objective function for projective clustering, taking into account the inherent trade-off between the dimension of a subspace and the induced clustering error. We then present an extension of the ¤ -means clustering algorithm for projective clustering in arbitrary subspaces, and also propose techniques to avoid local minima. Unlike previous algorithms, ours can choose the dimension of each cluster independently and automatically. Furthermore, experimental results show that our algorithm is significantly more accurate than the previous approaches.

We consider the problem of approximating a polygonal curve P under a given error criterion by another polygonal curve P whose vertices are a subset of the vertices of P. The goal is to minimize the number of vertices of P while ensuring that the error between P and P is below a certain threshold. We consider two different error measures: Hausdorff and Fréchet. For both error criteria, we present near-linear time approximation algorithms that, given a parameter ε > 0, compute a simplified polygonal curve P whose error is less than ε and size at most the size of an optimal simplified polygonal curve with error ε/2. We consider monotone curves in R 2 in the case of the Hausdorff error measure under the uniform distance metric and arbitrary curves in any dimension for the Fréchet error measure under L p metrics. We present experimental results demonstrating that our algorithms are simple and fast, and produce close to optimal simplifications in practice.

We consider the problem of computing minimum geometric hitting sets in which, given a set of geometric objects and a set of points, the goal is to compute the smallest subset of points that hit all geometric objects. The problem is known to be strongly NP-hard even for simple geometric objects like unit disks in the plane. Therefore, unless P=NP, it is not possible to get Fully Polynomial Time Approximation Algorithms (FPTAS) for such problems. We give the first PTAS for this problem when the geometric objects are half-spaces in R 3 and when they are an r-admissible set regions in the plane (this includes pseudo-disks as they are 2-admissible). Quite surprisingly, our algorithm is a very simple local search algorithm which iterates over local improvements only.

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We consider the problem of computing minimum geometric hitting sets in which, given a set of geometric objects and a set of points, the goal is to compute the smallest subset of points that hit all geometric objects. The problem is known to be strongly NP-hard even for simple geometric objects like unit disks in the plane. Therefore, unless P=NP, it is not possible to get Fully Polynomial Time Approximation Algorithms (FPTAS) for such problems. We give the first PTAS for this problem when the geometric objects are halfspaces in R 3 and when they are an r-admissible set regions in the plane (this includes pseudo-disks as they are 2-admissible). Quite surprisingly, our algorithm is a very simple local search algorithm which iterates over local improvements only.