CS133 Computational Geometry Convex Hull 1 Convex Hull Given a - PowerPoint PPT Presentation

CS133 Computational Geometry Convex Hull 1

Convex Hull Given a set of n points, find the minimal convex polygon that contains all the points 2

Convex Hull Properties θ 3

Convex Hull Representation The convex hull is represented by all its points sorted in CW/CCW order Special case: Three collinear points 4

Naïve Convex Hull Algorithm Iterate over all possible line segments A line segment is part of the convex hull if all other points are to its left Emit all segments in a CCW order Running time 𝑃 𝑜 3 5

Naïve Convex Hull Algorithm 6

Graham Scan Algorithm 7

Graham Scan Algorithm 5 11 3 7 4 10 8 6 14 2 1 13 12 9 15 0 8

Example 32

Graham Scan Pseudo Code Select the point with minimum 𝑧 Sort all points in CCW order 𝑞 0 , 𝑞 1 , … , 𝑞 𝑜 𝑇 = 𝑞 0 , 𝑞 1 For 𝑗 = 2 to 𝑜 While |𝑇| > 2 && 𝑞 𝑗 is to the right of 𝑇 −2 , 𝑇 −1 𝑇 .pop 𝑇 .push( 𝑞 𝑗 ) 33

Monotone Chain Algorithm Has some similarities with Graham scan algorithm Instead of sorting in CCW order, it sorts by one coordinate (e.g., x-coordinates) 34

Example 35

Pseudo Code Sort 𝑇 by 𝑦 𝑉 = {𝑇 0 } For 𝑗 = 1 to 𝑜 while |𝑉| > 1 && 𝑇 𝑗 is to the left of 𝑉 −2 𝑉 −1 𝑉 .pop 𝑉 .push( 𝑇 𝑗 ) 𝑀 = {𝑇 0 } While |𝑀| > 1 && 𝑇 𝑗 is to the right of 𝑀 −2 𝑀 −1 𝑀 .pop 𝑀 .push( 𝑇 𝑗 ) 36

Gift Wrapping Algorithm Start with a point on the convex hull Find more points on the hull one at a time Terminate when the first point is reached back Also knows as Jarvi’s March Algorithm 37

Gift Wrapping Example 38

Gift Wrapping Pseudo Code Gift Wrapping(S) CH = {} CH << Left most point do Start point = CH.last End point = CH[0] For each point s ∈ S If Start point = End Point OR s is to the left of 𝑇𝑢𝑏𝑠𝑢 𝑞𝑝𝑗𝑜𝑢, 𝐹𝑜𝑒 𝑞𝑝𝑗𝑜𝑢 End point = s CH << End point Running time 𝑃(𝑜 ⋅ 𝑙) 47

Divide & Conquer Convex Hull ConvexHull(S) Splits S into two subsets S1 and S2 Ch1 = ConvexHull(S1) Ch2 = ConvexHull(S2) Return Merge(Ch1, Ch2) 48

Divide & Conquer Convex Hull ConvexHull(S) Splits S into two subsets S1 and S2 Ch1 = ConvexHull(S1) Ch2 = ConvexHull(S2) Return Merge(Ch1, Ch2)   49

Merge: Upper Tangent 50

Merge: Lower Tangent 53

Merge Step Upper Tangent( 𝑀, 𝑆 ) 𝑄 𝑗 = Right most point in 𝑀 𝑄 𝑘 = Left most point in 𝑆 Do Done = true While 𝑄 𝑗+1 is to the right of 𝑞 𝑘 𝑞 𝑗 𝑗 + + ; done = false While 𝑄 𝑘−1 is to the left of 𝑞 𝑗 𝑞 𝑘 𝑘 − − ; done = false 59

Analysis Sort step: 𝑃 𝑜 ⋅ log 𝑜 Merge step: 𝑃 𝑜 Recursive part 𝑜 𝑈 𝑜 = 2𝑈 2 + 𝑑 ⋅ 𝑜 𝑈 𝑜 = 𝑃 𝑜 ⋅ log 𝑜 Overall running time 𝑃 𝑜 ⋅ log 𝑜 60

Incremental Convex Hull Start with an initial convex hull Add one additional point to the convex hull Given a convex hull CH and a point p, how to compute the convex hull of {CH, p}? Think: Insert an element into a sorted list 61

Case 1: p inside CH 62

Case 2: p on CH 63

Case 3: p outside CH 64

Case 3: p outside CH 65

Analysis of the Insert Function Test whether the point is inside, outside, or on the polygon O(n) Find the two tangents O(n) A more efficient algorithm can have an amortized running time of O(log n) 66

Quick Hull If we can have a divide-and-conquer algorithm similar to merge sort … why not having an algorithm similar to quick sort? Sketch Find a pivot Split the points along the pivot Recursively process each side 67

Quick Hull 68

Quick Hull How to split the points across the line segment? 69

Quick Hull How to select the farthest point? 70

Quick Hull 71

Quick Hull 72

Quick Hull How to split the points into three subsets? 73

Quick Hull 74

Quick Hull 75

Quick Hull 76

Quick Hull 77

Quick Hull 78

Quick Hull 79

Quick Hull 80

Example 81

Running Time Analysis 𝑈 𝑜 = 𝑈 𝑜 1 + 𝑈 𝑜 2 + 𝑃 𝑜 Worst case 𝑜 1 = 𝑜 − 𝑙 or 𝑜 2 = 𝑜 − 𝑙 , where 𝑙 is a small constant (e.g., k=1) 𝑈 𝑜 = 𝑃 𝑜 2 Best case 𝑜 1 = 𝑙 and 𝑜 2 = 𝑙 , where 𝑙 is a small constant In this case, most of the points are pruned 𝑈 𝑜 = 𝑃 𝑜 Average case, 𝑜 1 = 𝛽𝑜 and 𝑜 2 = 𝛾𝑜 , where 𝛽 < 1 and 𝛾 < 1 𝑈 𝑜 = 𝑃 𝑜 log 𝑜 82

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