Geometric algorithms for object placement and planarity in a terrain [Elektronische Ressource] / Rahul Ray
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Geometric algorithms for object placement and planarity in a terrain [Elektronische Ressource] / Rahul Ray

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Geometric Algorithms for Object Placement andPlanarity in a TerrainRahul Ray¨Max-Planck Institut fur InformatikGeometric Algorithms for Object Placement and Planarity in a TerrainGeometric Algorithms for Object Placement andPlanarity in a TerrainRahul RayDissertationzur Erlangung des GradesDoktor der Ingenieurwissenschaften (Dr.-Ing.)der Naturwissenschaftlich-Technischen Fakultat¨ Ider Universitat¨ des SaarlandesSaarbruck¨ enJuly 27th, 2004Max-Planck Institut fur¨ InformatikTag des Kolloquiums: 27 July 2004Dekan: Prof. Dr. Philipp SlusallekGutachter: Prof. Dr. Kurt MehlhornProf. Dr. Stefan SchirraDedicated to my parentsAcknowledgementsMany people have taught, inspired, encouraged, supported, helped and advised me during thetime in which I worked on this thesis. I wish to express my deepest gratitude to all of them,without which this thesis would never happen.First of all, I would like to thank my advisor, Prof. Dr. Kurt Mehlhorn, for providingme a perfect balance of scientific guidance and scientific freedom. He has always been a greatsource of motivation whenever I needed. I would like to thank him for creating a wonderfulresearch atmosphere in Max-Planck Institut fur¨ Informatik, Saarbruck¨ en. I also wish to thankDr. Stefan Funke and Dr. Theocharis Malamatos for co-guiding my research work that ledto this thesis. They are great persons to work with and I consider myself privileged to havecollaborated with them in my research.

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Published 01 January 2004
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Geometric Algorithms for Object Placement and Planarity in a Terrain
Rahul Ray
Max-PlanckInstitutf¨urInformatik
Geometric Algorithms for Object Placement and Planarity in a Terrain
Geometric Algorithms for Object Placement and Planarity in a Terrain
Rahul
Ray
Dissertation zur Erlangung des Grades Doktor der Ingenieurwissenschaften (Dr.-Ing.) der Naturwissenschaftlich-Technischen Fakulta¨t I der Universita¨t des Saarlandes
Saarbru¨cken July 27th, 2004
Max-PlanckInstitutfu¨rInformatik
Tag des Kolloquiums: Dekan: Gutachter:
27 July 2004 Prof. Dr. Philipp Slusallek Prof. Dr. Kurt Mehlhorn Prof. Dr. Stefan Schirra
Dedicated to my parents
Acknowledgements
Many people have taught, inspired, encouraged, supported, helped and advised me during the time in which I worked on this thesis. I wish to express my deepest gratitude to all of them, without which this thesis would never happen. First of all, I would like to thank my advisor,Prof. Dr. Kurt Mehlhorn, for providing me a perfect balance of scienticguidance and scienticfreedom. He has always been a great source of motivation whenever I needed. I would like to thank him for creating a wonderful research atmosphere in Max-Planck Institut fu¨r Informatik, Saarbru¨cken. I also wish to thank Dr. Stefan Funke and Dr. Theocharis Malamatos for co-guiding my research work that led to this thesis. They are great persons to work with and I consider myself privileged to have collaborated with them in my research. I share some of the most enjoyable moments in MPI with them which came along during many of our enlighting discussions or exchanges of innu-merable emails. They also helped me in proof-reading this thesis manuscript and improving the presentation a lot. When I go back to my uninteresting days in industry, I shall never forget Prof. Michiel Smid who gave me an opportunity to think about a PhD option. I am grateful to him for innite number of reasons. He introduced me to the planarity problem in terrain and many other interesting aspects of it while I was in Magdeburg University (Germany). I am thankful to ever smiling Prof. Ulrich Wendt for his wonderful collaboration from material science department in our project. I learned a lot from my co-authors, whose ideas and insightful suggestions were invaluable in discovering many of the algorithms presented in this thesis. My sincere thanks to them. I thank Prof. Stefan Schirra for kindly agreeing to review the thesis and to act as an examiner. I would like to thank and express my sincere gratitude to Suman Mukherjee, who pur-suaded me to continue this research at a time I thought to abandon it. My friend Arnab Paul has always kept on inspiring me even before I started my research. All my friends inTajmahal, Chhannachharahave been a constant source of inspiration for me to nishthis thesis one day. My special thanks to them. My MPII experience would not be the same without my friends, colleagues and ofce-mates (categories not mutually exclusive). I thank Naveen Sivadasan, Debapriyo, Hisao, Venkatesh, Tobias, Ingmar, Christian, Holgar, Kerstin, Petra, Arno, Nicola, Kavitha, Sunil and many others who are/were my colleagues in various times in MPII for their friendly help and making my life fun and enjoyable. I express my gratitude and thanks to Deb, Sudip, Chandrima, Pintu, Raja, Sai, Reddy, Venkat, Fawad, Tarang, Hareesh and many others in Saarcricket Club for providing wonderful socialinteractionwhichhastunredmystayinSaarbr¨uckenanunforgettableeventinmylife.
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Finally, I would like to thank those whom I owe the most. My parents have always en-couraged me to pursue higher studies even when I was in a kindergarten !! They have greatly inuenced my life with love and guidance and most importantly freedom of my soul which allowed me to reach where I am now. My then class mate and now wife, Piyali, was a great emotional support for me, whether it was a time for hard work or for taking it easy. To mark the submission of this thesis, we were blessed with a baby son, Rishav, on 22nd April. I gratefully acknowledge the fellowship fromInternational Max-Planck Research School (IMPRS)ha,tuptsrmatInfof¨uritutek.n¨rcuaabrkiS,charseremyedrtpotsnIkcnalP-xaMta
Abstract
We consider the followingplacement problem: LetCbe a compact set inR2and letSbe a set ofnpoints inR2. The objective is to compute a translate ofCthat contains the maximum number,, of points ofS. Motivated by the applications in clustering and pattern recognition, this optimal placement problem has received much attention over the last two decades. It is known that this problem can be solved in a time that is roughly quadratic inn. We show for a givenε >0and bucketing techniques can be used to develop ahow random-sampling near-linear-time Monte Carlo algorithm that computes a placement ofCcontaining at least (1 ε)points ofSwith high probability. WhenCan approximation algorithm for the optimal placementis a unit disk, we give problem by approximating the constraining radius of the disk. Here, our algorithm computes a placement of the disk of radius(1 +ε)which contains at leastpoints, wheredenotes the maximum number of points covered by any unit disk. The running time of this algorithm isO(n/ε2). Then, we turn to the problem of computing the largest connected region in a triangulated terrain of sizentriangles deviate by at most some small xedfor which the normals of the angle. We devise an exact algorithm by adapting dynamic graph connectivity algorithm to compute the largest planar region inO(n2logn(log logn)3)time. We also give a heuristic that can be used to compute sufcientlylarge planar regions in a terrain much faster. We discuss an implementation of this heuristic, and show some experimental results for terrains representing three-dimensional (topographical) images of fracture surfaces of metals obtained by confocal laser scanning microscopy, which directly motivated our research into this direction. Since the output of this heuristic comes with no quality assurance, we present a new ap-proximation scheme for the same problem which apart from giving a guarantee on the quality of the produced solution also has been implemented in practice and shows good performance on real-world data representing fracture surfaces. This approximate deterministic algorithm computes inO(n/ε2)time the largest approximately connected planar region in the terrain. We also give a variant of the above algorithm with a better dependence onεat the cost of an extra poly-logarithmic factor onn large a sufciently. Forn, both the algorithms consume optimalO(n)space.