CS133 Computational Geometry Convex Hull 4/12/2018 1 Convex Hull - PowerPoint PPT Presentation

CS133 Computational Geometry Convex Hull 4/12/2018 1

Convex Hull Given a set of n points, find the minimal convex polygon that contains all the points 4/12/2018 2

Convex Hull Properties θ 4/12/2018 3

Convex Hull Representation The convex hull is represented by all its points sorted in CW/CCW order Special case: Three collinear points 4/12/2018 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 4/12/2018 5

Naïve Convex Hull Algorithm 4/12/2018 6

Graham Scan Algorithm 4/12/2018 7

Graham Scan Algorithm 5 11 3 7 4 10 8 6 14 2 1 13 12 9 15 0 4/12/2018 8

Example 4/12/2018 32

Graham Scan Pseudo Code Select the point with minimum y Sort all points in CCW order {p 0 ,p 1 ,…, p n ) S = {p 0 ,p 1 } For i=2 to n While |S| > 2 && p i is to the right of S -2 ,S -1 S.pop S.push(p i ) 4/12/2018 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) 4/17/2018 34

Example 4/17/2018 35

Pseudo Code Sort S by x U = {S 0 } For i = 1 to n while |U| > 1 && S i is to the left of 𝑉 −2 𝑉 −1 U.pop U.push(S i ) L = {S 0 } While |L| > 1 && S i is to the right of 𝑀 −2 𝑀 −1 L.pop L.push(S i ) 4/17/2018 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 Jarvis March Algorithm 4/17/2018 37

Gift Wrapping Example 4/17/2018 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 O(n.k) 4/17/2018 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) 4/17/2018 48

Merge: Upper Tangent 4/17/2018 49

Merge: Lower Tangent 4/17/2018 52

Merge Step Upper Tangent(L,R) P i = Right most point in L P j = Left most point in R Do Done = true While P i+1 is to the right of 𝑞 𝑘 𝑞 𝑗 i++; done = false While P j-1 is to the left of 𝑞 𝑗 𝑞 𝑘 j--; done = false 4/17/2018 58

Analysis Sort step: O(n log n) Merge step: O(n) Recursive part T(n)=2T(n/2)+O(n) T(n)=O(n log n) Overall running time O(n log n) 4/17/2018 59

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 4/19/2018 60

Case 1: p inside CH 4/19/2018 61

Case 2: p on CH 4/19/2018 62

Case 3: p outside CH 4/19/2018 63

Case 3: p outside CH 4/19/2018 64

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) 4/19/2018 65

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 4/19/2018 66

Quick Hull 4/19/2018 67

Quick Hull How to split the points across the line segment? 4/19/2018 68

Quick Hull How to select the farthest point? 4/19/2018 69

Quick Hull 4/19/2018 70

Quick Hull 4/19/2018 71

Quick Hull How to split the points into three subsets? 4/19/2018 72

Quick Hull 4/19/2018 73

Quick Hull 4/19/2018 74

Quick Hull 4/19/2018 75

Quick Hull 4/19/2018 76

Quick Hull 4/19/2018 77

Quick Hull 4/19/2018 78

Quick Hull 4/19/2018 79

Example 4/19/2018 80

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 4/19/2018 81

CS133 Computational Geometry Convex Hull 4/12/2018 1 Convex Hull - PowerPoint PPT Presentation

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