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**1 - 9**of**9**### Approved by:

, 2004

"... E-Mail: piyush at acm dot org The thesis source is covered under the GNU General Public License. The thesis source comes with ABSOLUTELY NO WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. The sou ..."

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E-Mail: piyush at acm dot org The thesis source is covered under the GNU General Public License. The thesis source comes with ABSOLUTELY NO WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. The source is free software, and you are welcome to redistribute it under certain conditions. Click here for details. Printed on the Samsung CLP-500, Joe’s Press, NY. The paper used in this publication meets the minimum requirements of the Stony Brook Graduate School.

### Cluster Connecting Problem inside a Polygon

"... The cluster connecting problem inside a simple polygon is introduced in this paper. The problem is shown to be NP-complete. A log n-factor approximation algorithm is proposed. 1 ..."

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The cluster connecting problem inside a simple polygon is introduced in this paper. The problem is shown to be NP-complete. A log n-factor approximation algorithm is proposed. 1

### Abstract Locating Guards for Visibility Coverage of Polygons ∗

"... We propose heuristics for visibility coverage of a polygon with the fewest point guards. This optimal coverage problem, often called the “art gallery problem”, is known to be NP-hard, so most recent research has focused on heuristics and approximation methods. We evaluate our heuristics through expe ..."

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We propose heuristics for visibility coverage of a polygon with the fewest point guards. This optimal coverage problem, often called the “art gallery problem”, is known to be NP-hard, so most recent research has focused on heuristics and approximation methods. We evaluate our heuristics through experimentation, comparing the upper bounds on the optimal guard number given by our methods with computed lower bounds based on heuristics for placing a large number of visibility-independent “witness points”. We give experimental evidence that our heuristics perform well in practice, on a large suite of input data; often the heuristics give a provably optimal result, while in other cases there is only a small gap between the computed upper and lower bounds on the optimal guard number. 1

### Minimum Link Path Finding Methods in 2D Polygons

, 2007

"... The author surveys methods for finding a path between two arbitrary points within a 2D polygon under the link-distance metric. Early algorithms limited to simple polygons are explored first, and categorized into two different approaches. These methods are then evaluated for expansion into non-simple ..."

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The author surveys methods for finding a path between two arbitrary points within a 2D polygon under the link-distance metric. Early algorithms limited to simple polygons are explored first, and categorized into two different approaches. These methods are then evaluated for expansion into non-simple polygons (i.e. spaces with “Obstacles ” within the polygon boundary). Then, the problem of finding a minimum link path with obstacles is examined. Visibility based approaches are discussed, and similarities and optimizations made to algorithms for simple polygons are discussed. A critique of these methods based on bit-precision is noted, as well as related problems such as rectilinear variants, combined metrics (link and Euclidean distance), and pre-processing algorithms which charge a large amount of work to building an initial data structure, which then can perform arbitrary link distance queries quickly. 1.

### 3D Visibility Graph

"... The visibility graph is a fundamental geometric structure which is useful in many applications, including illumination and rendering, motion planning, pattern recognition, and sensor networks. While the concept of visibility graph is widely studied for 2D scenes, there is not any acceptable equivale ..."

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The visibility graph is a fundamental geometric structure which is useful in many applications, including illumination and rendering, motion planning, pattern recognition, and sensor networks. While the concept of visibility graph is widely studied for 2D scenes, there is not any acceptable equivalence of visibility graph for 3D space. In this paper we explain some reason for this absence. Then we try to find a new way to define geometric structure in 3D space. Following our new way, we easily define a new structure called 3D visibility graph which we believe is the natural way to extend visibility graph in 3D scenes. We show how to compute it in an acceptable time.

### Acknowledgments

, 2005

"... Katz was not simply an advisor who helped with the paper writing process. He was an advisor in the comprehensive sense of the word, providing advice and support both professionally and personally. He went above and beyond what was required of him, for example while on sabbatical he hosted me at his ..."

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Katz was not simply an advisor who helped with the paper writing process. He was an advisor in the comprehensive sense of the word, providing advice and support both professionally and personally. He went above and beyond what was required of him, for example while on sabbatical he hosted me at his place. One could not ask for a better advisor. Along the (PhD) way, many have been helpful to me and my work, in specific I would like to thank the following: Professor Joseph S. B. Mitchell whose outstanding knowledge of Computational Geometry and algorithms, together with his generous habit of sharing that knowledge, really opened up new fields of interest for me. Paz Carmi (yet another PhD student of Matya’s) who shares with me an affection (obsession) for terrains. From the mountain climbing in Thailand and the snowboarding in the high Alps ’ glaciers, to the theoretical aspects of terrain simplification, your sharp insights and suggestions were usually helpful, and always fun.

### A Heuristic Homotopic Path Simplification Algorithm

"... Abstract. We study the well-known problem of approximating a polygonal path P by a coarse one, whose vertices are a subset of the vertices of P. In this problem, for a given error, the goal is to find a path with the minimum number of vertices while preserving the homotopy in presence of a given set ..."

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Abstract. We study the well-known problem of approximating a polygonal path P by a coarse one, whose vertices are a subset of the vertices of P. In this problem, for a given error, the goal is to find a path with the minimum number of vertices while preserving the homotopy in presence of a given set of extra points in the plane. We present a heuristic method for homotopy-preserving simplification under any desired measure for general paths. Our algorithm for finding homotopic shortcuts runs in O(m log(n + m) +n log n log(nm)+k) time, where k is the number of homotopic shortcuts. Using this method, we obtain an O(n 2 + m log(n + m)+n log n log(nm)) time algorithm for simplification under the Hausdorff measure.