1. UMass Lowell Computer Science 91.504 Advanced Algorithms Computational Geometry Prof. Karen Daniels Spring, 2007 Lecture 2 Chapter 2: Polygon Partitioning Wednesday, 2/7/07 based on textbook Computational Geometry in C by Joseph O’Rourke

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 Polygon is monotone if it is monotone in neighborhood of every vertex ä Absence of interior cusps guarantees y-monotonicity ä Monotone polygon can be (greedily) triangulated in O(n) time ä deBerg et al.

4. Trapezoidalization ä Partition into trapezoids ä Horizontal line through each vertex ä Diagonal of each interior cusp with each “supporting” vertex yields monotone partition ä To trapezoidalize, vertically sweep horizontal 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 Triangulate each monotone polygon in O(n) time O(n lg n)

5. Partition into Monotone Mountains ä Alternative: ä Trapezoidalize (as before, time in O(nlgn)) ä 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) ä Connect each pair of trapezoid-supporting vertices that don’t lie on same (left/right) side of their trapezoid ä Triangulate mountains ä Total time in O(nlgn) (as before) Algorithm: TRIANGULATION of MONOTONE MOUNTAIN Identify base edge (find extreme points) Initialize internal angles at each nonbase vertex Link nonbase strictly convex vertices into a circular 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 see paper for details

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 segment query structure ä Coin-flip to decide which segment to add next ä log*n phases, each adding subset of segments in random order to query structure Trapezoidalize -> Monotone Mountain -> Triangulate see Section 7.11.4 for details

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: ä Hertel & Melhorn’s Algorithm: ä Essential diagonal d for vertex v if removing d creates nonconvex piece at v ä Iteratively remove inessential diagonals. ä Number of convex pieces: see paper Extra Credit: What algorithm does LEDA use?