1/60. Geometric Algorithms. Lecture 1: Introduction. Convex Hulls

Transcription

1 1/60 Geometric Algorithms Lecture 1: Introduction Convex Hulls

2 Geometric algorithms scope 2/60 Geometry algorithms (practice): Study of geometric problems that arise in various applications and how algorithms can help to solve well-defined versions of such problems Geometry algorithms (theory): Study of geometric problems on geometric data, and how efficient algorithms that solve them can be

3 Geometric problems examples 3/60 Geometric Networks,w e.g., are there relatively short routes between any two cities?

4 Geometric problems examples 3/60 Geometric Networks,w e.g., are there relatively short routes between any two cities? where do streets cross rivers?

5 Geometric problems examples 4/60 Planar Subdivisions, e.g., data structures for fast point location queries

6 Geometric problems examples 4/60 Planar Subdivisions, e.g., data structures for fast point location queries data structures for fast range queries

7 This course 5/60 lecturer: Kevin Buchin course website: http: // book: [BCKO] M. de Berg, O. Cheong, M. van Kreveld, M. Overmars, Computational Geometry: Algorithms and Applications (3rd edition). Springer-Verlag, Heidelberg, register on

8 This course: prerequisites 6/60 O-, Θ, Ω-notation; how to analyze algorithms Basic calculus: 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 Sorting algorithms Graph terminology, representations of graphs, basic graph algorithms Dynamic programming, greedy algorithms

9 This course 7/60 learning objectives: At the end of this course you should be able to... decide which algorithm or data structure to use in order to solve a given basic geometric problem, analyze new problems and come up with your own efficient solutions using concepts and techniques from the course.

10 This course 8/60 assessment: do 5 out of 6 homework assignments: 12% each on each assignment there is one exercise that you solve individually and two that you solve in a group do a course project (in groups of 3 students): 40% to pass you need 12.5 (out of 25) of points on your best 5 homework exercises done individually AND 5.5 as combined grade of assignments and project

11 This course: homework assignments 9/60 assignments once a week feel free to ask questions if an exercise is not clear to you if you want early feedback on your solution if you get stuck on an exercise Monday at lecture (best) or or in person (Mon/Tue (evening)/wed better make appointment)

12 This course: homework assignments 10/60 homework marked with a (G) is done as a project group homework not marked with a (G) is done individually & in particular the write-up needs to be done completely independent. If you discuss these exercises with other students, written notes on these discussions may not be used as basis for the homework.

13 This course: projects 11/60 choice between different experimentation projects, Option A: compute geometric spanners amidst obstacles (variation on Lecture 6 using Lecture 8) Option B: compute trapezoidal map of a simple polygon (Lecture 4 motivated by Lecture 3) you can also come up with your own project ideas (e.g., game or theory), and discuss them with me. find details also on course webpage 2IMA15/projects.html

14 This course: experimentation projects 12/60 General Guidelines and Report Writing take the guidelines on the web-page into careful consideration Data challenge groups contribute one data set to the data challenge of their problem groups run additionally to their regular experiments their algorithms on the data challenge sets Mini-presentations Groups give a 4-minute presentation at the end of the course presenting their results. This presentation is not graded. January 11 or 16?

15 This course: projects 13/60 Time line: asap form a group of your choice; if you would like me to help, contact me Nov 21 send with group members (and preferred topic) Dec 14 hand in a report outline and make an appointment to discuss it with me (Dec 19/20/21) Jan 2 send data set for data challenge (if applicable) Jan 11 or 16 project presentations Jan 18 hand in deliverables

16 Geometry: points, lines,... 14/60 Plane (two-dimensional), R 2 Space (three-dimensional), R 3 Space (higher-dimensional), R d A point in the plane, 3-dimensional space, higher-dimensional space. p = (p x, p y ), p = (p x, p y, p z ), p = (p 1, p 2,..., p d ) 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...

17 Geometry: line segments 15/60 A line segment pq is defined by its two endpoints p and q: (λ p x + (1 λ) q x, λ p y + (1 λ) q y ) 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

18 Polygons: simple or not 16/60 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

19 Other shapes: rectangles, circles, disks 17/60 A circle is only the boundary, a disk is the boundary plus the interior Rectangles, squares, quadrants, slabs, half-lines,...

20 Relations: distance, intersection, angle 18/60 The distance between two points is generally the Euclidean distance: (px q x ) 2 + (p y q y ) 2 (px q x ) 2 + (p y q y ) 2 ) Another option: the Manhattan distance: p x q x + p y q y The distance between two geometric objects other than points usually refers to the minimum distance between two points that are part of these objects p x q x p y q y

21 Relations: distance, intersection, angle 19/60 The intersection of two geometric objects is the set of points (part of the plane, space) they have in common Question 1: What is the maximum number of intersection points of a line and a simple polygon with 10 vertices (trick question)?

22 Relations: distance, intersection, angle 20/60 The intersection of two geometric objects is the set of points (part of the plane, space) they have in common Question 2: What is the maximum number of intersection points of a line and a simple polygon boundary with 10 vertices (still a trick question)?

23 Description size 21/60 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) 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 y = m x + c false, m, c x = c true,.., c

24 Description size and Computation time 22/60 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 We assume: Any computation (distance, intersection) on two objects of O(1) description size takes O(1) time!

25 Algorithms, efficiency 23/60 Recall from your algorithms and data structures course: A set of n real numbers can be sorted in O(nlogn) 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!

26 Wildlife Analysis 24/ Black Rhino (Diceros bicornis) Extent of Occurrence

27 Convexity 25/60 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?

28 Convex hull 26/60 For any subset of the plane (set of points, rectangle, simple polygon), its convex hull is the smallest convex set that contains that subset

29 Convex hull problem 27/60 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?

30 Convex hull problem 28/60 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?

31 Developing an algorithm 29/60 To develop an algorithm, find useful properties, make various observations, draw many sketches to gain insight

32 Developing an algorithm 30/60 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 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 p q

33 Developing an algorithm 31/60 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 pq to E 8. From the set E of edges construct a list L of vertices of CH(P), sorted in clockwise order.

34 Developing an algorithm 32/60 Question: How must line 5 be interpreted to make the algorithm correct, i.e., how do we handle degeneracies? Question: How efficient is the algorithm? Question: Is the algorithm robust against rounding errors? Note: Robustness is a huge issue when implementing geometric algorithms

35 Developing an algorithm 33/60 Another approach: incremental, from left to right Let s first compute the upper boundary of the convex hull this way (property: on the upper hull, points appear in x-order) Main idea: Sort the points from left to right (= by x-coordinate). Then insert the points in this order, and maintain the upper hull so far

36 Developing an algorithm 34/60 Observation: from left to right, there are only right turns on the upper hull

37 Developing an algorithm 35/60 Initialize by inserting the leftmost two points

38 Developing an algorithm 36/60 If we add the third point there will be a right turn at the previous point, so we add it

39 Developing an algorithm 37/60 If we add the fourth point we get a left turn at the third point

40 Developing an algorithm 38/60... so we remove the third point from the upper hull when we add the fourth

41 Developing an algorithm 39/60 If we add the fifth point we get a left turn at the fourth point

42 Developing an algorithm 40/60... so we remove the fourth point when we add the fifth

43 Developing an algorithm 41/60 If we add the sixth point we get a right turn at the fifth point, so we just add it

44 Developing an algorithm 42/60 We also just add the seventh point

45 Developing an algorithm 43/60 When adding the eight point... we must remove the seventh point

46 Developing an algorithm 44/60... we must remove the seventh point

47 Developing an algorithm 45/60... and also the sixth point

48 Developing an algorithm 46/60... and also the fifth point

49 Developing an algorithm 47/60 After two more steps we get:

50 The pseudo-code 48/60 Algorithm CONVEXHULL(P) Input. A set P of points in the plane. Output. A list containing the vertices of CH(P) in clockwise order. 1. Sort the points by x-coordinate, resulting in a sequence p 1,..., p n. 2. Put the points p 1 and p 2 in a list L upper, with p 1 as the first point. 3. for i 3 to n 4. do Append p i to L upper. 5. while L upper contains more than two points and the last three points in L upper do not make a right turn 6. do Delete the middle of the last three points from L upper.

51 The pseudo-code 49/60 p 1, p 2, p 10, p 13, p 14 Then we do the same for the lower convex hull, from right to left We remove the first and last points of the lower convex hull... and concatenate the two lists into one p 14, p 12, p 8, p 4, p 1

52 Algorithm analysis 50/60 Algorithm analysis generally has two components: proof of correctness efficiency analysis, proof of running time

53 Correctness 51/60 Are the general observations on which the algorithm is based correct? Does the algorithm handle degenerate cases correctly? Here: Does the sorted order matter if two or more points have the same x-coordinate? What happens if there are three or more collinear points, in particular on the convex hull?

54 Efficiency 52/60 Identify of each line of pseudo-code how much time it takes, if it is executed once (note: operations on a constant number of constant-size objects take constant time) Consider the loop-structure and examine how often each line of pseudo-code is executed Sometimes there are global arguments why an algorithm is more efficient than it seems, at first

55 The pseudo-code 53/60 Algorithm CONVEXHULL(P) Input. A set P of points in the plane. Output. A list containing the vertices of CH(P) in clockwise order. 1. Sort the points by x-coordinate, resulting in a sequence p 1,..., p n. 2. Put the points p 1 and p 2 in a list L upper, with p 1 as the first point. 3. for i 3 to n 4. do Append p i to L upper. 5. while L upper contains more than two points and the last three points in L upper do not make a right turn 6. do Delete the middle of the last three points from L upper.

56 Efficiency 54/60 The sorting step takes O(nlogn) time Adding a point takes O(1) time for the adding-part. Removing points takes constant time for each removed point. If due to an addition, k points are removed, the step takes O(1 + k) time Total time: O(nlogn) + n i=3 O(1 + k i ) if k i points are removed when adding p i Since k i = O(n), we get O(nlogn) + n i=3 O(n) = O(n 2 )

57 Efficiency 55/60 Global argument: each point can be removed only once from the upper hull This gives us the fact: Hence, n i=3 k i n O(nlogn) + n i=3 O(1 + k i ) = O(nlogn) + O(n) = O(nlogn)

58 Result 56/60 The convex hull of a set of n points in the plane can be computed in O(nlogn) time, and this is worst-case optimal.

59 Result 56/60 The convex hull of a set of n points in the plane can be computed in O(nlogn) time, and this is worst-case optimal. Can we do better?

60 Result 56/60 The convex hull of a set of n points in the plane can be computed in O(nlogn) time, and this is worst-case optimal. Can we do better? O(nh) if the CH has h vertices? O(nlogh)? Yes we can.

61 Other approaches: divide-and-conquer 57/60 Divide-and-conquer: split the point set in two halves, compute the convex hulls recursively, and merge A merge involves finding extreme vertices in every direction

62 Other approaches: divide-and-conquer 58/60 Alternatively: split the point set in two halves on x-coordinate, compute the convex hulls recursively, and merge A merge now comes down to finding two common tangent lines

63 Convex hulls in 3D 59/60 For a 3-dimensional point set, the convex hull is a convex polyhedron It has vertices (0-dim.), edges (1-dim.), and facets (2-dim.) in its boundary, and a 3-dimensional interior The boundary is a planar graph, so it has O(n) vertices, edges and facets

64 Convex hulls in 4D 60/60 For a 4-dimensional point set, the convex hull is a convex polyhedron It has vertices (0-dim.), edges (1-dim.), 2-facets (2-dim.), and 3-facets (3-dim.) in its boundary, and a 4-dimensional interior Its boundary can have Θ(n 2 ) facets in the worst case!