UMass Lowell Computer Science Advanced Algorithms Computational Geometry Prof. Karen Daniels Spring, 2004 Lecture 2 Chapter 2: Polygon Partitioning Monday, 2/2/04

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

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

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 Triangulate each monotone polygon in O(n) time O(n lg n)

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)

Linear-Time Triangulation YearComplexityAuthors

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

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

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: