1. UMass Lowell Computer Science 91.504 Advanced Algorithms Computational Geometry Prof. Karen Daniels Spring, 2001 Lecture 2 Chapter 2: Polygon Partitioning Chapter 3: 2D Convex Hulls Monday, 2/12/01

2. Chapter 2 Polygon Partitioning useful for triangulations and many other uses!

3. Monotone Partitioning ä A chain is monotone with respect to a line L if every line orthogonal to L intersects the chain in at most 1 point ä P is monotone with respect to a line L if boundary of P can be split into 2 polygonal chains A and B such that each chain is monotone with respect to L ä Monotonicity implies sorted order with respect to L ä Monotone polygon can be (greedily) triangulated in O(n) time

4. Trapezoidalization ä Partition into trapezoids ä Horizontal line through each vertex ä Diagonal with each “supporting” vertex yields monotone partition ä To trapezoidalize, vertically sweep a line L ä presort vertices by y (O(nlogn) time) ä maintain sorted list of edges intersecting L ä lg n lookup/insert/delete time (e.g. ht-balanced tree) ä for each vertex ä find edge left and right along sweep line Algorithm: POLYGON TRIANGULATION: MONOTONE PARTITION Sort vertices by y coordinate Perform plane sweep to construct trapezoidalization Partition into monotone polygons by connecting from interior cusps Triangulation each monotone polygon in O(n) time O(n lg n)

5. Partition into Monotone Mountains ä One monotone chain is a single segment ä Every strictly convex vertex is an ear tip (except maybe base endpoints) Algorithm: TRIANGULATION of MONOTONE MOUNTAIN Identify base edge Initialize internal angles at each nonbase vertex Link nonbase strictly convex vertices into a list while list nonempty do For convex vertex b, remove triangle abc Output diagonal ac Update angles and list O(n)

6. Linear-Time Triangulation YearComplexityAuthors

7. Linear-Time Triangulation ä Chazelle’s Algorithm (High-Level Sketch) ä Computes visibility map ä horizontal chords left and right from each vertex ä Algorithm is like MergeSort (divide-and-conquer) ä Partition polygon of n vertices into n/2 vertex chains ä Merge visibility maps of subchains to get one for chain ä Improve this by dividing process into 2 phases: 1) Coarse approximations of visibility maps for linear- time merge 2) Refine coarse map into detailed map in linear time

8. Seidel’s Randomized Triangulation ä Simple, practical algorithm ä Randomized: Coin-flip for some decisions ä Build trapezoidalization quickly ä O(log n) expected cost for locating point in query structure ä Coin-flip to decide which segment to add next Trapezoidalize -> Monotone Mountain -> Triangulate

9. Convex Partitioning ä Competing Goals: ä minimize number of convex pieces ä minimize partitioning time ä To add points or not add points? Theorem (Chazelle): Let  be the fewest number of convex pieces into which a polygon may be partitioned. For a polygon of r reflex vertices:

10. Chapter 3 2D Convex Hulls

11. Convexity & Convex Hulls  A convex combination of points x 1,..., x k is a sum of the form  1 x 1 +...+  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

12. 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 )

13. 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

14. 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)

15. 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) 

16. 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

17. Incremental Algorithm 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)

18. 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

19. Homework 1 Fri, 2/9 Wed, 2/14 Chapter 1 (O’Rourke) problems 1 & 2 problems 1 & 2 Extra credit may be turned in any time during week of 2/12 2 Mon, 2/12 Wed, 2/21 Chapters 2,3 (O’Rourke) problems 1 & 2 Extra credit may be turned in any time during week of 2/21 HW# Assigned Due Content

20. Machine Accounts ä Each student has an account on my machine: minkowski.cs.uml.edu ä Username is the same as your username on CS ä Password is your initials followed by the last 5 digits on the bottom right hand corner of the back of your student id card (1 exception) ä To remotely log in, use a secure shell (e.g. ssh) ä To transfer files, use a secure FTP (e.g. sftpc) ä Saturn has an sftpc client ä Code to use with assignments will be there.

21. Preparing for Homework 2 ä Graham Scan 2D Convex Hull Code ä Background on research problem that motivates problem 2