UMass Lowell Computer Science Analysis of Algorithms Prof. Karen Daniels Spring, 2002 Tuesday, 5/7/02 Computational Geometry Chapter 33

Relevant Sections of Chapter 35 Ch33 Computational Geometry You’re responsible for material in this chapter that we discuss in lecture. (Note that this includes all sections.)

Overview ä Computational Geometry Introduction ä Line Segment Intersection ä Convex Hull Algorithms ä Nearest Neighbors/Closest Points

Computational Geometry Telecommunications Visualization & Bioinformatics Manufacturing ComputerGraphics Design Analyze Apply

Typical Problems ä bin packing ä Voronoi diagram ä simplifying polygons ä shape similarity ä convex hull ä maintaining line arrangements ä polygon partitioning ä nearest neighbor search ä kd-trees SOURCE: Steve Skiena’s Algorithm Design Manual (for problem descriptions, see graphics gallery at ) (for problem descriptions, see graphics gallery at

Common Computational Geometry Structures Voronoi Diagram Delaunay Triangulation Convex Hull New Point source: O’Rourke, Computational Geometry in C

Sample Tools of the Trade Algorithm Design Patterns/Techniques: binary searchdivide-and-conquerduality randomizationsweep-line derandomizationparallelism Algorithm Analysis Techniques: asymptotic analysis, amortized analysis Data Structures: winged-edge, quad-edge, range tree, kd-tree Theoretical Computer Science principles: NP-completeness, hardness Growth of Functions Summations Recurrences Sets Probability MATH Proofs Geometry Graph Theory Combinatorics Linear Algebra

Computational Geometry in Context TheoreticalComputerScience Applied Computer Science AppliedMath Geometry ComputationalGeometryEfficient Geometric Algorithms Design Analyze Apply

Line Segment Intersections (2D) Intersection of 2 Line Segments Intersection of > 2Line Segments

Cross-Product-Based Geometric Primitives source: textbook Cormen et al. p0p0p0p0 p2p2p2p2 p1p1p1p1 (1) p1p1p1p1 p3p3p3p3 p2p2p2p2 (2) p2p2p2p2 p1p1p1p1 p3p3p3p3 p4p4p4p4 (3) Some fundamental geometric questions:

Cross-Product-Based Geometric Primitives: (1) source: textbook Cormen et al. Advantage: less sensitive to accumulated round-off error p0p0p0p0 p2p2p2p2 p1p1p1p1 (1) 33.1

Cross-Product-Based Geometric Primitives: (2) source: textbook Cormen et al. p1p1p1p1 p3p3p3p3 p2p2p2p2 (2) 33.2

Intersection of 2 Line Segments source: textbook Cormen et al. p2p2p2p2 p1p1p1p1 p3p3p3p3 p4p4p4p4 (3) Step 1: Bounding Box Test Step 2: Does each segment straddle the line containing the other? 33.3

Segment-Segment Intersection ä Finding the actual intersection point ä Approach: parametric vs. slope/intercept ä parametric generalizes to more complex intersections ä Parameterize each segment a b c d L ab L cd a b c d L ab L cd A=b-a p(s)=a+sA Intersection: values of s, t such that p(s) =q(t) : a+sA=c+tC q(t)=c+tC C=d-c source: O’Rourke, Computational Geometry in C

Demo Segment/Segment Intersection

Intersection of >2 Line Segments source: textbook Cormen et al. Sweep-Line Algorithmic Paradigm: 33.4

Intersection of >2 Line Segments source: textbook Cormen et al. Sweep-Line Algorithmic Paradigm:

Intersection of >2 Line Segments source: textbook Cormen et al. O(n lg n) Time to detect if any 2 segments intersect: source: textbook Cormen et al. 33.5

Intersection of Segments ä Goal: “Output-size sensitive” line segment intersection algorithm that actually computes intersection points ä Bentley-Ottmann plane sweep: O((n+k)logn) time ä k = number of intersection points in output ä Intuition: sweep horizontal line downwards ä just before intersection, 2 segments are adjacent in sweep-line intersection structure ä check for intersection only segments that adjacent ä event types: ä top endpoint of a segment ä bottom endpoint of a segment ä intersection between 2 segments Improved to O(nlogn+k) [Chazelle/Edelsbrunner] source: O’Rourke, Computational Geometry in C

Convex Hull Algorithms Definitions Gift Wrapping Graham Scan QuickHullIncrementalDivide-and-Conquer Lower Bound in  (nlgn)

Convexity & Convex Hulls  A convex combination of points x 1,..., x k is a sum of the form  1 x  k x k where ä Convex hull of a set of points is the set of all convex combinations of points in the set. nonconvex polygon convex hull of a point set source: O’Rourke, Computational Geometry in C source: textbook Cormen et al.

Naive Algorithms for Extreme Points Algorithm: INTERIOR POINTS for each i do for each j = i do for each j = i do for each k = j = i do for each k = j = i do for each L = k = j = i do for each L = k = j = i do if p L in triangle(p i, p j, p k ) then p L is nonextreme O(n 4 ) Algorithm: EXTREME EDGES for each i do for each j = i do for each j = i do for each k = j = i do for each k = j = i do if p k is not left or on (p i, p j ) if p k is not left or on (p i, p j ) then (p i, p j ) is not extreme O(n 3 ) source: O’Rourke, Computational Geometry in C

Algorithms: 2D Gift Wrapping ä Use one extreme edge as an anchor for finding the next  O(n 2 ) Algorithm: GIFT WRAPPING i 0 index of the lowest point i i 0 repeat for each j = i for each j = i Compute counterclockwise angle  from previous hull edge Compute counterclockwise angle  from previous hull edge k index of point with smallest  k index of point with smallest  Output (p i, p k ) as a hull edge Output (p i, p k ) as a hull edge i k i k until i = i 0 source: O’Rourke, Computational Geometry in C

Gift Wrapping source: textbook Cormen et al. 33.9

Algorithms: 3D Gift Wrapping CxHull Animations: O(n 2 ) time [output sensitive: O(nF) for F faces on hull]

Algorithms: 2D QuickHull ä Concentrate on points close to hull boundary ä Named for similarity to Quicksort a b O(n 2 ) Algorithm: QUICK HULL function QuickHull(a,b,S) if S = 0 return() if S = 0 return() else else c index of point with max distance from ab c index of point with max distance from ab A points strictly right of (a,c) A points strictly right of (a,c) B points strictly right of (c,b) B points strictly right of (c,b) return QuickHull(a,c,A) + (c) + QuickHull(c,b,B) return QuickHull(a,c,A) + (c) + QuickHull(c,b,B) source: O’Rourke, Computational Geometry in C

Algorithms: 3D QuickHull CxHull Animations:

Algorithms: Qhull (>= 2D )

Graham’s Algorithm ä Points sorted angularly provide “star-shaped” starting point ä Prevent “dents” as you go via convexity testing O(nlgn) Algorithm: GRAHAM SCAN, Version B Find rightmost lowest point; label it p 0. Sort all other points angularly about p 0. In case of tie, delete point(s) closer to p 0. In case of tie, delete point(s) closer to p 0. Stack S (p 1, p 0 ) = (p t, p t-1 ); t indexes top i 2 while i < n do if p i is strictly left of p t-1 p t then Push(p i, S) and set i i +1 else Pop(S)  source: O’Rourke, Computational Geometry in C

Graham Scan source: textbook Cormen et al.

Graham Scan source: textbook Cormen et al. 33.7

Graham Scan source: textbook Cormen et al. 33.7

Graham Scan source: textbook Cormen et al.

Graham Scan source: textbook Cormen et al.

Algorithms: 2D Incremental ä Add points, one at a time ä update hull for each new point ä Key step becomes adding a single point to an existing hull. ä Idea is extended to 3D in Chapter 4. O(n 2 ) Algorithm: INCREMENTAL ALGORITHM Let H 2 ConvexHull{p 0, p 1, p 2 } for k 3 to n - 1 do H k ConvexHull{ H k-1 U p k } H k ConvexHull{ H k-1 U p k } can be improved to O(nlgn) source: O’Rourke, Computational Geometry in C

Algorithms: 3D Incremental CxHull Animations: O(n 2 ) time

Algorithms: 2D Divide-and-Conquer ä Divide-and-Conquer in a geometric setting ä O(n) merge step is the challenge ä Find upper and lower tangents ä Lower tangent: find rightmost pt of A & leftmost pt of B; then “walk it downwards” ä Idea is extended to 3D in Chapter 4. Algorithm: DIVIDE-and-CONQUER Sort points by x coordinate Divide points into 2 sets A and B: A contains left n/2 points B contains right n/2 points Compute ConvexHull(A) and ConvexHull(B) recursively Merge ConvexHull(A) and ConvexHull(B) O(nlgn) A B source: O’Rourke, Computational Geometry in C

Algorithms: 3D Divide and Conquer CxHull Animations: O(n log n) time !

Lower Bound of O(nlgn)  Worst-case time to find convex hull of n points in algebraic decision tree model is in  (nlgn) ä Proof uses sorting reduction: ä Given unsorted list of n numbers: (x 1,x 2,…, x n ) ä Form unsorted set of points: (x i, x i 2 ) for each x i ä Convex hull of points produces sorted list! ä Parabola: every point is on convex hull ä Reduction is O(n) (which is o(nlgn))  Finding convex hull of n points is therefore at least as hard as sorting n points, so worst-case time is in  (nlgn) Parabola for sorting 2,1,3 source: O’Rourke, Computational Geometry in C

Nearest Neighbor/ Closest Pair of Points

Closest Pair source: textbook Cormen et al. Goal: Given n (2D) points in a set Q, find the closest pair under the Euclidean metric in O(n lgn) time. Divide-and-Conquer Strategy: - -X = points sorted by increasing x - -Y = points sorted by increasing y - -Divide: partition with vertical line L into P L, P R - -Conquer: recursively find closest pair in P L, P R - -  L,  R are closest-pair distances - -  min(  L,  R ) - -Combine: closest-pair is either  or pair straddles partition line L - - Check for pair straddling partition line L - - both points must be within  of L - - create array Y’ = Y without points in 2  strip - - for each point p in Y’ - - find (<= 7) points in Y’ within  of p

Closest Pair source: textbook Cormen et al. Correctness 33.11

Closest Pair source: textbook Cormen et al. Running Time: Key Point: Presort points, then at each step form sorted subset of sorted array in linear time