Computational Geometry Algorithmische Geometrie

Transcription

1 Algorithmische Geometrie Panos Giannopoulos Wolfgang Mulzer Lena Schlipf AG TI SS 2013

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3 Outline What you need to know (before taking this course) What is the course about? Course information Basics Convex Hulls I Assignment 1 FU Berlin SS

4 You need to know (or learn quickly) Basic geometry O Ω Θ-notation; how to analyze algorithms Basic discrete maths: manipulating summations solving recurrences working with logarithms etc. Basic probability theory: events probability distributions random variables expected values etc. Basic data structures: linked lists stacks queues Balanced binary search trees (AVL trees etc.) Sorting algorithms Basic graph theory (terminology algorithms etc.) Basic algorithmic paradigms: divide & conquer greedy algorithms dynamic programming etc. FU Berlin SS

5 This course is about Computational geometry (theory): Study of geometric problems on geometric data and how efficient algorithms that solve them can be Computational geometry (practice): Study of geometric problems that arise in various applications and how algorithms can help to solve well-defined versions of such problems FU Berlin SS

6 More on convex hulls Combinatorial complexity Computational geometry This course is about Computational geometry theory smallest enclosing circle Computational geometry (theory): Classify abstract geometric problems into classes depending on how efficiently they can be solved closest pair any intersection? find all intersections FU Berlin SS

7 This course is about: Example I Intersections Arrangements Point location intersection point Tasks: compute all object intersections Arrangement face compute and maintain the induced arrangement (points edges faces) do point location: find the face that a query point lies in query point Efficiency is important! FU Berlin SS

8 This course is about: Example II Robot motion planning Finding a shortest path amidst obstacles 13 we saw how to plan a path for a robot from a given start position goal position. The algorithm we gave always finds a path if it exists de no claims about the quality of the path: it could make a large make lots of unnecessary turns. In practical situations we would Tasks: nd not just any path but a good path. find a collision-free path for the robot from start-position to target-position collision-free: the robot does not intersect any ostacle find a shortest such path several versions: 2d 3d translating/rotating robots etc. Figure 15.1 A shortest path Efficiency is important! onstitutes a good path depends on the robot. In general the longer a ore time it will take the robot to reach its goal position. For a mobile factory floor this means it can transport less goods per time unit a loss of productivity. Therefore we would prefer a short path. Often her issues that play a role as well. For example some robots can only straight line; they have to slow down stop and rotate before they FU Berlin SS 2013 moving into a different direction so any turn along the path causes 8

9 Introduction Geometric objects Geometric relations Convex hulls Combinatorial complexity Computational geometry More on convex hulls applications Computational geometry Computational geometry practice Application areas that require geometric algorithms are computer graphics motion planning and robotics geographic information systems CAD/CAM statistics physics simulations databases games multimedia retrieval... Computing shadows from virtual light sources Spatial interpolation from groundwater pollution measurements Computing a collision-free path between obstacles Computing similarity of two shapes for shape database retrieval FU Berlin SS

10 Course information Lectures: Panos Giannopoulos (April half of May July English) Wolfgang Mulzer (half of May June July English/Deutsch) Be present in the lectures! Tutorials: Lena Schlipf Be present in the tutorials! (Especially if you have difficulties solving the exercises) Homework assignements: you may solve them in groups of two (Programming?) Homework assignements: you can hand in the solutions in English/Deutsch Homework assignements due: Monday before the Tutorial starts (10 days) There will be a written exam! In order to pass the course: 60% of the assignments in total 20% of each assignment (you can skip at most one) present two exercises in the tutorials (dance as well?) pass the exam FU Berlin SS

11 Course information (cont.) Course webpage: Book: M. de Berg O. Cheong M. van Kreveld M. Overmars. : Algorithms and Applications. Springer (3rd ed.). FU Berlin SS

12 More on convex hulls Combinatorial complexity Computational geometry Geometry: Points lines... Geometry: points lines... Plane (two-dimensional) R2 Space (three-dimensional) R3 Space (higher-dimensional) Rd A point in the plane 3-dimensional space higher-dimensional space. p = (px py ) p = (px py pz ) p = (p1 p2... pd ) A line in the plane: y = m x + c; representation by m and c A half-plane in the plane: y m x + c or y m x + c Represent vertical lines? Not by m and c... FU Berlin SS

13 Introduction Convex hulls More on convex hulls Geometry: line segments Geometric relations Combinatorial complexity Computational geometry Geometry: line segments A line segment pq is defined by its two endpoints p and q: (λ px + (1 λ ) qx λ py + (1 λ ) qy ) where 0 λ 1 Line segments are assumed to be closed = with endpoints not open Two line segments intersect if they have some point in common. It is a proper intersection if it is exactly one interior point of each line segment FU Berlin SS

14 More on convex hulls Combinatorial complexity Computational geometry Polygons: simple or not Polygons: simple or not A polygon is a connected region of the plane bounded by a sequence of line segments interior exterior simple polygon polygon with holes convex polygon non-simple polygon The line segments of a polygon are called its edges the endpoints of those edges are the vertices Some abuse: polygon is only boundary or interior plus boundary FU Berlin SS

15 Convex hulls More on convex hulls Rectangles circles disks... Combinatorial complexity Computational geometry Other shapes: rectangles circles disks A circle is only the boundary a disk is the boundary plus the interior Rectangles squares quadrants slabs half-lines wedges... FU Berlin SS

16 Convex hulls More on convex hulls Relations: Combinatorial complexity Computational geometry distance intersection angle Relations: distance intersection angle The distance between two points is generally the Euclidean distance: p (px qx )2 + (py qy )2 p (px qx )2 + (py qy )2 ) Another option: the Manhattan distance: px qx + py qy Question: What is the set of points at equal Manhattan distance to some point? py qy px qx FU Berlin SS

17 Convex hulls More on convex hulls Relations: Combinatorial complexity Computational geometry distance intersection angle Relations: distance intersection angle The distance between two geometric objects other than points usually refers to the minimum distance between two points that are part of these objects Question: How can the distance between two line segments be realized? FU Berlin SS

18 Convex hulls More on convex hulls Relations: Combinatorial complexity Computational geometry distance intersection angle Relations: distance intersection angle The intersection of two geometric objects is the set of points (part of the plane space) they have in common Question 1: How many intersection points can a line and a circle have? Question 2: What are the possible outcomes of the intersection of a rectangle and a quadrant? FU Berlin SS

19 Convex hulls More on convex hulls Relations: Combinatorial complexity Computational geometry distance intersection angle Relations: distance intersection angle Question 3: What is the maximum number of intersection points of a line and a simple polygon with 10 vertices (trick question)? FU Berlin SS

20 Convex hulls More on convex hulls Relations: Combinatorial complexity Computational geometry distance intersection angle Relations: distance intersection angle Question 4: What is the maximum number of intersection points of a line and a simple polygon boundary with 10 vertices (still a trick question)? FU Berlin SS

21 Convex hulls More on convex hulls Relations: Combinatorial complexity Computational geometry distance intersection angle Relations: distance intersection angle Question 5: What is the maximum number of edges of a simple polygon boundary with 10 vertices that a line can intersect? FU Berlin SS

22 Description size Convex hulls More on convex hulls Combinatorial complexity Computational geometry Description size y =m x+c A point in the plane can be represented using two reals A line in the plane can be represented using two reals and a Boolean (for example) false m c A line segment can be represented by two points so four reals A circle (or disk) requires three reals to store it (center radius) A rectangle requires four reals to store it x=c true.. c FU Berlin SS

23 Description size Convex hulls More on convex hulls Combinatorial complexity Computational geometry Description size A simple polygon in the plane can be represented using 2n reals if it has n vertices (and necessarily n edges) A set of n points requires 2n reals A set of n line segments requires 4n reals A point line circle... requires O(1) or constant storage. A simple polygon with n vertices requires O(n) or linear storage FU Berlin SS

24 Computation time Any computation (distance intersection) on two objects of O(1) description size takes O(1) time! (Practice: detecting intersections and computing intersections can be different) Theory: Real RAM i.e. all usual arithmetic operations can be done in constant time (+ / sin log...) FU Berlin SS

25 Convex hulls More on convex hulls Algorithms efficiency Combinatorial complexity Computational geometry Algorithms efficiency Recall from your algorithms and data structures course: A set of n real numbers can be sorted in O(n log n) time A set of n real numbers can be stored in a data structure that uses O(n) storage and that allows searching insertion and deletion in O(log n) time per operation These are fundamental results in 1-dimensional computational geometry! FU Berlin SS

26 Convex hulls More on convex hulls Convexity Algorithm development Algorithm analysis Convexity A shape or set is convex if for any two points that are part of the shape the whole connecting line segment is also part of the shape Question: Which of the following shapes are convex? Point line segment line circle disk quadrant? FU Berlin SS

27 More on convex hulls Algorithm analysis Convex Convexhull hull For any subset of the plane (set of points rectangle simple polygon) its convex hull is the smallest convex set that contains that subset Demo FU Berlin SS

28 Algorithm analysis Convexhull hullproblem problem Convex Give an algorithm that computes the convex hull of any given set of n points in the plane efficiently The input has 2n coordinates so O(n) size Question: Why can t we expect to do any better than O(n) time? FU Berlin SS

29 Convex hull problem Convex hull problem Assume the n points are distinct The output has at least 4 and at most 2n coordinates so it has size between O(1) and O(n) The output is a convex polygon so it should be returned as a sorted sequence of the points counterclockwise along the boundary Question: Is there any hope of finding an O(n) time algorithm? FU Berlin SS

30 Convex hull problem Convex hull problem Assume the n points are distinct The output has at least 4 and at most 2n coordinates so it has size between O(1) and O(n) The output is a convex polygon so it should be returned as a sorted sequence of the points counterclockwise along the boundary Question: Is there any hope of finding an O(n) time algorithm? Lower bound: Any convex hull algorithm requires Ω(n log n) operations in the quadratic decision tree model [Yao 1981] FU Berlin SS

31 Convex hull problem Convex hull problem Assume the n points are distinct The output has at least 4 and at most 2n coordinates so it has size between O(1) and O(n) The output is a convex polygon so it should be returned as a sorted sequence of the points counterclockwise along the boundary Question: Is there any hope of finding an O(n) time algorithm? Lower bound: Any convex hull algorithm requires Ω(n log n) operations in the quadratic decision tree model [Yao 1981] Computational 1: Introduction and Convex Hulls However: Perhaps wegeometry can getlecture an output sensitive algorithm that runs in O(n log h) time where h is the size of the hull! FU Berlin SS

32 Convex hulls More on convex hulls Developing an algorithm Algorithm development Algorithm analysis Developing an algorithm To develop an algorithm find useful properties make various observations draw many sketches to gain insight Property: The vertices of the convex hull are always points from the input Consequently the edges of the convex hull connect two points of the input Property: The supporting line of any convex hull edge has all input points to one side q p all points lie left of the directed line from p to q if the edge from p to q is a CCW convex hull edge FU Berlin SS

33 Convex hulls More on convex hulls Developing an algorithm Algorithm development Algorithm analysis Developing an algorithm Algorithm SlowConvexHull(P) Input. A set P of points in the plane. Output. A list L containing the vertices of CH(P) in clockwise order. 1. E 0. / 2. for all ordered pairs (p q) P P with p not equal to q 3. do valid true 4. for all points r P not equal to p or q 5. do if r lies left of the directed line from p to q 6. then valid false 7. if valid then Add the directed edge p~q to E 8. From the set E of edges construct a list L of vertices of CH(P) sorted in clockwise order. FU Berlin SS

34 Convex hulls More on convex hulls Developing an algorithm Algorithm development Algorithm analysis Developing an algorithm Question: How must line 5 be interpreted to make the algorithm correct? Question: How efficient is the algorithm? FU Berlin SS

35 at or m es if ve ts st of he We of ss w is nd be nd rs at to he Section 1.1 Degeneracies AN EXAMPLE : CONVEX HULLS 3 points of P could lie on a line Correct criterion for convex hull edge: edge pq is an edge of CH(P) iff all other points r P lie either strictly to the right of the directed line through p q or on the open line segment pq (assuming no coinciding points in P) At first when developing an algorithm you can assume general position... q r p FU Berlin SS

36 Geometry in Theory Left of Right of... Robustness px orientation(p q r ) = sign det qx rx py qy ry 1 < 0 cw(p q r ) 1 = = 0 colinear(p q r ) 1 > 0 ccw(p q r ) = sign((qx px )(ry py ) (qy py )(rx px )) ccw(p q s) ccw(s q r ) ccw(p s r ) ccw(p q r ) p r s q Practice: rounding errors in floating point arithmetic can give 6 Algorithm Library wrong result of the predicate. We won t care about this here! FU Berlin SS

37 Running time? FU Berlin SS

38 Running time? O(n3 ) We didn t use any clever ideas! Friday: O(n log n) and O(n log h) algorithms (h: size of the output) FU Berlin SS

39 Assignment 1 http: //page.mi.fu-berlin.de/panos/cg13/exercises/u01.pdf Due: 15/04 before the Tutorial starts! FU Berlin SS