Computational Geometry. Algorithm Design (10) Computational Geometry. Convex Hull. Areas in Computational Geometry

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1 Computational Geometry Algorithm Design (10) Computational Geometry Graduate School of Engineering Takashi Chikayama Algorithms formulated as geometry problems Broad application areas Computer Graphics, Geographic Information system (GIS), Robotics, CAD/CAM, Numerical Control (NC), A variety of (seemingly unrelated) application areas formulated as geometry problems Geometry in feature space for representing objects Applicable when the number of dimensions is relatively low 2 Areas in Computational Geometry Two types of problems Numerical/machine geometry Computer-aided geometric design Geometric modeling Continuous numerical models for real-world objects Computational/algorithmic geometry Deals with discrete models Serves as the foundation of numerical/machine geometry Convex Hull The minimum convex figure that contains all points of the given set of points Can be uniquely represented by a set of vertices (they are always convex polytopes or polygons) Can be defined in non-euclidian spaces also 3 4 1

2 Gift Wrapping Algorithm Also known as Jarvis s march (R. A. Jarvis 1973) 1. Select the leftmost point A, which is a point in the convex hull 2. Select point B such that the angle between line AB is the steepest 3. Let B be the new A and go back to 2 B A 5 Properties of the Gift Wrapping algorithm Time complexity To find the first point: O(n) To find the next point, you need to compute the angle for every remaining point O(n) Number of iterations: h (number of vertices of the convex hull) In total: O(hn) Not too bad when h is small Can be naturally extended to spaces with more than two dimensions 6 Graham Scan Properties of Graham Scan Robert Graham, 1972 To find the upper half of the convex hull 1. Sort the points according to their x-coordinates 2. Let P be the leftmost point RQ P 3. Let Q be the next point QR 4. Let R be the next point P Q 5. Compare the angles of PQ and PR i.e. check the sign of the cross product of the two 6. If PR is upper, then let R be the new Q and go back to 4 7. Otherwise, Q is a point in the convex hull. Let Q be the new P and go back to 3 7 Time complexity Sorting: O(n log n) Cost of comparing angles is constant Total cost for comparison: O(n) Sorting is dominant: O(n log n) Whether this is better than gift wrapping (O(hn)) or not depends on n and h Cannot be easily applied to many-dimensional cases 8 2

3 Divide and Conquer algorithm Compute the convex hulls for the subsets of the points and integrate them into larger sets 1. Sort the points according to x-coordinates 2. Create triangles of every consecutive 3 points (they are all convex hulls of 3 points) 3. Integrate them by building bridges between adjacent convex hulls How to build bridges 1. Given two convex hulls A and B, let P be the rightmost point in A and Q be the leftmost point in B 2. If the next point R in A is above the line PQ, then let R be the new P and the next point be the new R, and repeat this process 3. Do the same for Q 4. Go back to 2 and repeat until no new selection is made A B 9 P Q 10 Properties of the Divide and Conquer algorithm Time complexity Sorting: O(n log n) Creating triangles: O(n) Integrating convex hulls of size m: O(m) Merging adjacent convex hulls: O(n) log n steps of merging are needed: O(n log n) In total: O(n log n) Equal to that of Graham scan Can be naturally extended to multi-dimensions Amenable to parallelization (common feature of 11 Divide & conquer) Kirkpatrick-Seidel algorithm D. G. Kirkpatrick and R. Seidel 1983 Improves on Div+conq: O(n log n) O(n log h) Build bridges before creating convex hulls Marriage before conquest The points below the bridge can be discarded 12 3

4 Akl-Toussaint heuristics Maximal elements S. Akl and G. T. Toussaint, 1978 Ignore points that cannot be a vertex of the convex hull Find the points with max/min x/y-coordinates All the points that lie in the tetragon defined by the four points can be ignored Time complexity: O(n) Optionally, you can use an octagon which can be built in a similar manner 13 Given two points in a k- dimensional space, we define a partial order p q as p is not less than q in any dimension Maximal elements M(S) of S is defined as the set of points such that, for any point p in M(S), there exists no such x in S that x p 14 Two-Dimensional Case: Plane Sweeping Sort the elements of S according to x-coordinates p 1, p 2,, p n p n is a maximal element its x-coordinate is maximum Walk through the points p n-1, p n-2,, p 1 and select the points that have the maximum y-coordinates so far Sorting is dominant: O(n log n) No maximal elements here 15 Maximal Elements in High Dimensions Sort the points according to x-coordinates: p 1, p 2,, p n p n is a maximal element If p k is a maximal element in S k = { p k, p k+1,, p n }, then it is also a maximal element in S The x-coordinates of p k is greater than that of any points { p 1, p 2,, p k 1 } Judging whether or not p k is a maximal element in S k Project S k to a plane perpendicular to the x-axis If it is a maximal element on the plane, then it is a maximal element in S k p k is not less than another point in some axis Solving a problem with one fewer dimension 16 4

5 Finding a Maximal Element by Projection y x z The point with the maximum x-coordinate is a maximal element Project the points to the y z plane The maximal elements on the y z plane are maximal elements in the original space 17 Closest Pair Problem Given n points in a space, find a pair of points with the smallest distance between them Naively compare every pair of points: O(n 2 ) One-dimensional case: sort & compare O(n log n) Not applicable to multi-dimensional cases x x 18 Divide and Conquer 1. Sort the points according to their x-coordinates 2. Divide them into two at some x 3. Compute the closest pair in each group recursively 4. Merge the results from the two groups Merging Operation Let 1 and 2 be the distances of the closest pairs of the two sets The distance of the closest pair after the merging operation is = min( 1, 2 ) or the distance between two points (p and q) belonging to the two different sets The distances of p and q from the boundary must not be greater than 1 d(p, q) d(p, q) <?

6 How Many Points to Check? Merging: Time Complexity For every point whose distance to the boundary is not greater than, you need to check the inside of the rectangle (W =, H = 2) on the other side We already know that the distance between any points in the rectangle is not smaller than 6 points at most (2-D) 2 3 k-1 points at most (k-d) 21 Detecting the points close to the boundary O(n) Two-dimensional case Check the distances along the y-axis For each point, you need to check 6 points at most O(n) This algorithm can be easily extended to multidimensional cases: O(n) Cost of one merging operation is proportional to the number of points to be merged log n steps of merging operations are needed to 22 integrate all points O(n log n) in total Voronoi Diagram Decompose a space according to the distance to the nearest point in the given set of points S V(p i ) = { x p j S d(x, p i ) d(x, p j ) } Applications: finding the nearest post office, etc. Plane Sweep Algorithm (Fortune s algorithm) Steven Fortune 1986 Sort the points according to their x-coordinates Sweep a vertical line L t (i.e. x = t) from left to right, decomposing the left side of the line according to the distance to the nearest point or L t Lines constituting the Voronoi diagram L t Parabola curves on the frontier 23 t x 24 6

7 Plane Sweep algorithm (cont d) Events on the sweep line L t L t crosses a point i.e. t = x(p i ) Add a new parabola curve v Three parabola curves cross an intersection point of two lines The two line segments end when t = v x + d(v,p), where d(v,p) is the distance to the nearest points You need to maintain each event point t using a priority queue Time complexity: O(n log n) 25 Variants of Voronoi Diagrams Furthest Point Voronoi Diagrams Identify the areas that are furthest from the points Manhattan Voronoi Diagrams Use Manhattan distance Weighted Voronoi Diagrams Assign weight w s to each point and use weighted distances (s, p) = d(s, p)/w s (s, p) = d(s, p) 2 w s 2 etc. 26 Triangulation Delaunay Triangulation Fill a polygon with triangles Not uniquely determined even if the vertices to be used are given High dimensions: Fill a polytope with simplexes Tetrahedrons in three dimensions No circumscribed circles of the triangles contain any external points The dual structure of the Voronoi diagram

8 Properties of Delaunay Triangulation Maximizes the minimum angle of all the angles of the triangles If a circle whose diameter is defined by two of the points does not contain any other point, then the edge connecting the two is included in the Delaunay triangulation Applications of Delaunay Triangulation Efficient construction of minimum spanning trees in Euclidean spaces (described later) Modeling the surface of an object E.g. representing a curved surface by triangulation in computer graphics + shading realistic pictures Resulting models are smooth because the minimum angle is maximized Creating the grid for the finite element method How to Construct a Delaunay Triangulation Divide and conquer: similar to the closest pair problem Divide the given points into two sets Construct the Delaunay triangulation for each Merge them Apply this process recursively The merging operation costs O(n), so the total cost is O(n log n) Euclidean Minimum Spanning Tree (EMST) Build a minimum spanning tree (MST) for n points on a plane MST problem with a complete graph Standard MST algorithms are not efficient because the graph has O(n 2 ) edges Euclidean distance allows a more efficient solution 1. Construct a Delaunay triangulation: O(n log n) The resulting graph has 3n edges at most because it is a graph on a plane 2. Apply a standard MST algorithm O(n log n)

9 EMST is a Part of the Delaunay Triangulation A) Cycle Property: the longest edge of a cycle is not contained in a MST (proved later) B) Delaunay triangulation: if a circle whose diameter is defined by two of the points does not contain any other point, then the edge connecting the two is included in the Delaunay triangulation p A circle whose diameter is not included in the Delaunay r triangulation as an edge must contain another point This contradicts with (A) because q pq is longer than pr and qr 33 pq is not included in the MST Cycle Property of MST The longest edge of a cycle is not contained in a MST The tree can be divided into two by removing it The two trees can be reconnected by another edge which was not present in the original tree The resulting tree is shorter than the original tree 34 Other Problems: Geometric Optimization Minimum Diameter Spanning Tree Minimize the maximum distance between any two points in the tree Minimum Enclosing Circle Compute the minimum circle that encloses all elements of a set of points Largest Empty Circle/Rectangle No points must lie in the circle/rectangle Shortest Path Can be solved more efficiently than in the general setting if you can assume Euclid distance 35 Other Problems: Constructive Solid Geometry Intersection Detection Compute whether two objects have a common part or not Constructive Solid Geometry Union Intersection Difference Computer graphics Industrial applications: CAD, Collision detection, etc. 36 9

10 CSG Examples Union Intersection Difference by User Captain Sprite on en.wikipedia 37 Created by POV-Ray (open-source ray tracing software) 38 10