Panagiotis D. Alevizos

The New k-Windows Algorithm for Improving thek -Means Clustering Algorithm

The process of partitioning a large set of patterns into disjoint and homogeneous clusters is fundamental in knowledge acquisition. It is called Clustering in the literature and it is applied in various fields including data mining, statistical data analysis, compression and vector quantization. The k-means is a very popular algorithm and one of the best for implementing the clustering process. The k-means has a time complexity that is dominated by the product of the number of patterns, the number of clusters, and the number of iterations. Also, it often converges to a local minimum. In this paper, we present an improvement of the k-means clustering algorithm, aiming at a better time complexity and partitioning accuracy. Our approach reduces the number of patterns that need to be examined for similarity, in each iteration, using a windowing technique. The latter is based on well known spatial data structures, namely the range tree, that allows fast range searches.

Parallelizing the Unsupervised k-Windows Clustering Algorithm

Clustering can be defined as the process of partitioning a set of patterns into disjoint and homogeneous meaningful groups, called clusters. The growing need for parallel clustering algorithms is attributed to the huge size of databases that is common nowadays. This paper presents a parallel version of a recently proposed algorithm that has the ability to scale very well in parallel environments mainly regarding space requirements but also gaining a time speedup.

An optimal O ( n log n ) algorithm for contour reconstruction from rays

We present an optimal algorithm to reconstruct the planar cross section of a simple object from data points measured by rays. The rays are semi-infinite curves representing, for example, the laser beam or the articulated arms of a robot moving around the object. The object is assumed to be a unique simply connected object, and the contour to be reconstructed is a simple polygon having the data points as vertices and intersecting none of the measuring rays. Such a contour does not exist for any given sets of points and rays but only for legal data. In this paper, we prove that the solution to the contour problem is unique whenever such a solution exists. For a set of n points and n rays, the algorithm presented here provides in &Ogr;(nlogn) time, a polygon which is the solution to the contour problem when the data are legal. Updating this contour if a new measure is available can be done in &Ogr;(logn) time. Both results are asymptotically optimal in the worst-case. Moreover, once the solution has been found, we can check if the data are legal in &Ogr;(nlogn) time.

Improving the orthogonal range search k-windows algorithm

Clustering, that is the partitioning of a set of patterns into disjoint and homogeneous meaningful groups (clusters), is a fundamental process in the practice of science. k-windows is an efficient clustering algorithm that reduces the number of patterns that need to be examined for similarity. using a windowing technique. It exploits well known spatial data structures, namely the range free, that allows fast range searches. From a theoretical standpoint, the k-windows algorithm is characterized by lower time complexity compared to other well-known clustering algorithms. Moreover it achieves high quality clustering results. However, it appears that it cannot be directly applicable in high-dimensional settings due to the superlinear space requirements for the range tree. In this paper an improvement of the k-windows algorithm, aiming at resolving this deficiency, is presented. The improvement is based on an alternative solution to the orthogonal range search problem.

Non-convex contour reconstruction

We present algorithms to reconstruct the planar cross-section of a simply connected object from data points measured by rays. The rays are semi-infinite curves representing, for example, the laser beam or the articulated arms of a robot moving around the object. This paper shows that the information provided by the rays is crucial (though generally neglected) when solving 2-dimensional reconstruction problems. The main property of the rays is that they induce a total order on the measured points. This order is shown to be computable in optimal time O(n log n). The algorithm is fully dynamic and allows the insertion or the deletion of a point in O(log n) time. From this order a polygonal approximation of the object can be deduced in a straightforward manner. However, if insufficient data are available or if the points belong to several connected objects, this polygonal approximation may not be a simple polygon or may intersect the rays. This can be checked in O(n log n) time. The order induced by the rays can also be used to find a strategy for discovering the exact shape of a simple (but not necessarily convex) polygon by means of a minimal number of probes. When each probe outcome consists of a contact point, a ray measuring that point and the normal to the object at the point, we have shown that 3n-3 probes are necessary and sufficient if the object has n non-colinear edges. Each probe can be determined in O(log n) time yielding an O(n log n)-time 0(n)-space algorithm. When each probe outcome consists of a contact point and a ray measuring that point but not the normal, the same strategy can still be applied. Under a mild condition, 8n-4 probes are sufficient to discover a shape that is almost surely the actual shape of the object.

Parallel Unsupervised k-Windows: An Efficient Parallel Clustering Algorithm

Clustering can be defined as the process of partitioning a set of patterns into disjoint and homogeneous meaningful groups (clusters). There is a growing need for parallel algorithms in this field since databases of huge size are common nowadays. This paper presents a parallel version of a recently proposed algorithm that has the ability to scale very well in parallel environments.

On the order induced by a set of rays: application to the probing of nonconvex polygons

The authors present a strategy for discovering the exact shape of a simple (but not necessarily convex) polygon by means of a minimal number of simple probes. When each probe outcome consists of a contact point, a ray measuring that point and the normal to the object at the point, it is shown that 3n-3 probes are necessary and sufficient to discover the exact shape of a polygon with n noncollinear edges. Each probe can be determined in O(log n) time, yielding on O(n log n)-time O(n)-space algorithm.<<ETX>>

An optimal algorithm for the boundary of a cell in a union of rays

In this paper, we study a cell of the subdivision induced by a union of n half lines (or rays) in the plane. We present two results. The first one is a novel proof of the O(n) bound on the number of edges of the boundary of such a cell, which is essentially of methodological interest. The second is an algorithm for constructing the boundary of any cell, which runs in optimal Θ(n log n) time. A byproduct of our results are the notions of skeleton and of skeletal order, which may be of interest in their own right.

Pictures as Boolean Formulas

Both labellability and realizability problems of planar projections of polyhedra (i.e., pictures) are known to be NP-complete problems. This is true, even in the case of trihedral polyhedra, where exactly three faces meet at every vertex. In this paper, we examine pictures that are taken to be projections of trihedral polyhedra without holes, and contain the projections of all edges (hidden and visible) of a polyhedron. In other words, we examine pictures which represent the entire shape of a trihedral polyhedron without holes. Such a picture is a connected graph P=(V,E) with |E| edges and |V| nodes, each of degree 3 (

Improving the Orthogonal Range Search k-windows Clustering Algorithm

|E| = \frac{3|V|}{2}