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11
Automatic Subspace Clustering of High Dimensional Data
 Data Mining and Knowledge Discovery
, 2005
"... Data mining applications place special requirements on clustering algorithms including: the ability to find clusters embedded in subspaces of high dimensional data, scalability, enduser comprehensibility of the results, nonpresumption of any canonical data distribution, and insensitivity to the or ..."
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Cited by 687 (12 self)
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Data mining applications place special requirements on clustering algorithms including: the ability to find clusters embedded in subspaces of high dimensional data, scalability, enduser comprehensibility of the results, nonpresumption of any canonical data distribution, and insensitivity to the order of input records. We present CLIQUE, a clustering algorithm that satisfies each of these requirements. CLIQUE identifies dense clusters in subspaces of maximum dimensionality. It generates cluster descriptions in the form of DNF expressions that are minimized for ease of comprehension. It produces identical results irrespective of the order in which input records are presented and does not presume any specific mathematical form for data distribution. Through experiments, we show that CLIQUE efficiently finds accurate clusters in large high dimensional datasets.
Covering Rectilinear Polygons with AxisParallel Rectangles
, 1999
"... We give an O(sqrt(log n)) factor approximation algorithm for covering a rectilinear polygon with holes using axisparallel rectangles. This is the first polynomial time approximation algorithm for this problem with a o(log n) approximation factor. ..."
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Cited by 26 (1 self)
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We give an O(sqrt(log n)) factor approximation algorithm for covering a rectilinear polygon with holes using axisparallel rectangles. This is the first polynomial time approximation algorithm for this problem with a o(log n) approximation factor.
Close Approximations of Minimum Rectangular Coverings
 In FST & TCS'96, volume 1180 of LNCS
, 1996
"... . We consider the problem of covering arbitrary polygons with rectangles. The rectangles must lie entirely within the polygon. (This requires that the interior angles of the polygon are all greater than or equal to 90 degrees.) We want to cover the polygon with as few rectangles as possible. Thi ..."
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Cited by 7 (2 self)
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. We consider the problem of covering arbitrary polygons with rectangles. The rectangles must lie entirely within the polygon. (This requires that the interior angles of the polygon are all greater than or equal to 90 degrees.) We want to cover the polygon with as few rectangles as possible. This problem has an application in fabricating masks for integrated circuits. In this paper we will describe the first polynomial algorithm, guaranteeing an O(log n) approximation factor, provided that the n vertices of the input polygon are given as polynomially bounded integer coordinates. By the same technique we also obtain the first algorithm producing a covering which is within a constant factor of the optimal in exponential time (compared to the doublyexponential known before). 1 Introduction The problem of covering polygons with various types of simpler polygons has a number of important practical applications [5, 6] and has received considerable attention from a theoretical p...
Applying Partial Evaluation to VLSI Design Rule Checking
, 1995
"... This report describes the design and implementation of a complete VLSI design rule checking program. We use formal techniques to develop a methodology for performing design rule checking, and implement this methodology in the Scheme programming language. We specify the requirements for a simplified ..."
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Cited by 2 (0 self)
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This report describes the design and implementation of a complete VLSI design rule checking program. We use formal techniques to develop a methodology for performing design rule checking, and implement this methodology in the Scheme programming language. We specify the requirements for a simplified VLSI design database and implement it, making use of the Hilbert Rtree. We describe the implementationof an efficient algorithm for the decompositionof rectilinear polygons into collections of rectangles. We apply partial evaluation techniques to our final design rule checking program in order to determine the effect this has on our program's structure and performance. Finally, we describe the implementation of a graphical user interface for the checker and summarise our experiences and insights gained during the course of this project.
Two Geometric Optimization Problems
 in DingZhu Du and Jie Sun (eds.), New Advances in Optimization and Approximation
, 1994
"... Abstract. We consider two optimization problems with geometric structures. The rst one concerns the following minimization problem, termed as the rectilinear polygon cover problem: \Cover certain features of a given rectilinear polygon (possibly with rectilinear holes) with the minimum number of rec ..."
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Abstract. We consider two optimization problems with geometric structures. The rst one concerns the following minimization problem, termed as the rectilinear polygon cover problem: \Cover certain features of a given rectilinear polygon (possibly with rectilinear holes) with the minimum number of rectangles included in the polygon. " Depending upon whether one wants to cover the interior, boundary or corners of the polygon, the problem is termed as the interior, boundary or corner cover problem, respectively. Most of these problems are known to be NPcomplete. In this chapter we survey some of the important previous results for these problems and provide a proof of impossibility of a polynomialtime approximationscheme for the interior and boundary cover problems. The second problem concerns routing in a segmented routing channel. The related problems are fundamental to routing and design automation for Field Programmable Gate Arrays (FPGAs), anewtype of electrically programmable VLSI. In this chapter we survey the theoretical results on the combinatorial complexity and algorithm design for segmented channel routing. It is known that the segmented channel routing problem is in general NPComplete. E cient polynomial time algorithms for a number of important special cases are presented.
Efficient Storage Compression for 3D Regions
"... In this work we present heuristics algorithms for efficient storage compression for 3D regions. Giving a 3D decomposed in parallelepipeds, we want to store it using the least storage information. We present the steps and the experimental results of five algorithms; the first one is a simple, space c ..."
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In this work we present heuristics algorithms for efficient storage compression for 3D regions. Giving a 3D decomposed in parallelepipeds, we want to store it using the least storage information. We present the steps and the experimental results of five algorithms; the first one is a simple, space consuming, approach that works as the upper bound for the storage requirements of the other four algorithms. In brief we present the steps of the algorithm of FranzblauKleitman (Franzblau and Kleitman 1984). This algorithm has a very good average performance. The algorithm works well for 2D regions. We produced an invariant of their algorithm for 3D regions. Our contribution is the development of the other three algorithms that have less storage requirements than the algorithm of FranzblauKleitman. Using experimental testing procedures (the results are presented in the conclusion) we present the evaluation of these algorithms. From this evaluation the fifth algorithm gives the best performance in every case. This algorithm behaves better for every case compared with the algorithm of (Franzblau and Kleitman 1984) with a near to optimal performance. 1.
unknown title
, 1991
"... Abstract We show the following results for polygons without holes: 1. covering the interior or boundary of an arbitrary polygon with convex polygons is NPhard, 2. covering the vertices of an arbitrary polygon with convex polygons is NPcomplete, 3. covering the interior or boundary of an orthogonal ..."
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Abstract We show the following results for polygons without holes: 1. covering the interior or boundary of an arbitrary polygon with convex polygons is NPhard, 2. covering the vertices of an arbitrary polygon with convex polygons is NPcomplete, 3. covering the interior or boundary of an orthogonal polygon with rectangles is NPcomplete. We note that these results hold even if the polygons are required to be in general position.
Geometric Decompositions and Networks  Approximation Bounds and Algorithms
"... In this thesis we focus on four problems in computational geometry: In the first four chapters we consider the problem of covering an arbitrary polygon with simpler polygons, i.e., rectangles. We present several approximation algorithms for this problem, and also some lower bounds on the number of ..."
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In this thesis we focus on four problems in computational geometry: In the first four chapters we consider the problem of covering an arbitrary polygon with simpler polygons, i.e., rectangles. We present several approximation algorithms for this problem, and also some lower bounds on the number of rectangles needed in a covering of a holefree polygon and on the timecomplexity for this and related problems. Then, we consider a generalization of the wellknown Euclidean traveling salesman problem (TSP), namely the TSP with neighborhoods problem. In the TSP with neighborhoods problem we are given a collection of polygonal regions, and we seek the shortest tour that visits each neighborhood at least once. We give approximation algorithms for the problem and also show a result on the hardness of the problem. Next we turn our attention to the problem of finding a tspanner of a complete geometric graph. The aim is to produce a sparse graph with a small number of edges and with low total weight, that is almost as &quot;good &quot; as a complete graph. With good we mean that for every pair of points in the graph there exists a path in the spanner graph that is at most t times longer than the distance between the two points. We present several approximation algorithms for this problem. In the final chapter of the thesis we introduce the concept of higherorder Delaunay triangulations. We give an algorithm to compute which edges can be included in a higherorder Delaunay triangulation. We show that for 1 order Delaunay triangulations, most of the criteria we study can be optimized in O(n log n) time, for example, minimizing the number of local minima, the number of local extrema, the maximum angle, area triangle, and degree of any vertex.
Close Approximations of Minimum Rectangular Coverings ∗
, 1998
"... Abstract. We consider the problem of covering arbitrary polygons with rectangles. The rectangles must lie entirely within the polygon. (This requires that the interior angles of the polygon are all greater than or equal to 90 degrees.) We want to cover the polygon with as few rectangles as possible. ..."
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Abstract. We consider the problem of covering arbitrary polygons with rectangles. The rectangles must lie entirely within the polygon. (This requires that the interior angles of the polygon are all greater than or equal to 90 degrees.) We want to cover the polygon with as few rectangles as possible. This problem has an application in fabricating masks for integrated circuits. In this paper we will describe the first polynomial algorithm, guaranteeing an O(log n) approximation factor, provided that the n vertices of the input polygon are given as polynomially bounded integer coordinates. By the same technique we also obtain the first algorithm producing a covering which is within a constant factor of the optimal in exponential time (compared to the doublyexponential known before).
COVERING RECTILINEAR POLYGONS WITH AXISPARALLEL RECTANGLES ∗
"... Abstract. We give an O ( log n) factor approximation algorithm for covering a rectilinear polygon with holes using axisparallel rectangles. This is the first polynomial time approximation algorithm for this problem with an o(log n) approximation factor. ..."
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Abstract. We give an O ( log n) factor approximation algorithm for covering a rectilinear polygon with holes using axisparallel rectangles. This is the first polynomial time approximation algorithm for this problem with an o(log n) approximation factor.