Graduate Algorithms CS F-19 Computational Geometry

Transcription

1 Graduate Algorithms CS F-19 Comutational Geometry David Galles Deartment of Comuter Science University of San Francisco

2 19-0: Cross Products Given any two oints 1 = (x 1,y 1 ) and = (x,y ) Cross Product: 1 = x 1 y x y 1 1 = x 1 y x y 1 = 1 (x y 1 x 1 y ) = 1

3 19-1: Cross Products Cross Product 1 as Signed Area Area is ositive if 1 is below Area is negative if 1 is above

4 19-: Cross Products Given two vectors that share an origin: 0 1 and 0 Is 0 clockwise or counterclockwise relative to 0?

5 19-3: Cross Products Counterclockwise 0 Clockwise

6 19-4: Cross Products Given two vectors that share an origin: 0 1 and 0 Is 0 clockwise or counterclockwise relative to 0? ( 1 0 ) ( 0 ) is ositive, 0 is counterclockwise from 0 1

7 19-5: Cross Products Given two line segments 0 1 and 1, which direction does angle 0 1 turn? Left Turn Right Turn

8 19-6: Cross Products Given two line segments 0 1 and 1, which direction does angle 0 1 turn? ( 0 ) ( 1 0 ) is ositive, left turn ( 0 ) ( 1 0 ) is negative, right turn ( 0 ) ( 1 0 ) is zero, no turn (colinear)

9 19-7: Line Segment Intersection Given two line segments 1 and 3 4, do they intersect? How could we determine this?

10 19-8: Line Segment Intersection Given two line segments 1 and 3 4, do they intersect? Each segment straddles the line containing the other An endoint of one segment lies on the other segment

11 19-9: Line Segment Intersection and 4 straddle line defined by 1 and 1 and straddle line defined by 3 and 4

12 19-10: Line Segment Intersection and 4 straddle line defined by 1 and 1 and do not straddle line defined by 3 and 4

13 19-11: Line Segment Intersection and 4 do not straddle line defined by 1 and 1 and do not straddle line defined by 3 and 4

14 19-1: Line Segment Intersection 3 4 1

15 19-13: Line Segment Intersection Counterclockwise Clockwise

16 19-14: Line Segment Intersection Clockwise Clockwise

17 19-15: Line Segment Intersection 3 and 4 straddle line define by 1 and if: 1 3 is counterclockwise of 1 and 1 4 is clockwise of 1 ( 1 ) ( 3 1 ) > 0 and ( 1 ) ( 4 1 ) < is clockwise of 1 and 1 4 is counterclockwise of 1 ( 1 ) ( 3 1 ) < 0 and ( 1 ) ( 4 1 ) > 0

18 19-16: Line Segment Intersection How can we determine if 3 is on the segment 1?

19 19-17: Line Segment Intersection How can we determine if 3 is on the segment 1? 3 is on the line defined by 1 and 3 is in the roer range along that line

20 19-18: Line Segment Intersection 3 1

21 19-19: Line Segment Intersection 3 1

22 19-0: Line Segment Intersection How can we determine if 3 is on the segment 1? 3 is on the line defined by 1 and ( 1 ) ( 3 1 ) = 0 3 is in the roer range along that line 3 x 1 x&& 3 x x or 3 x 1 x&& 3 x x 3 y 1 y&& 3 y y or 3 y 1 y&& 3 y y

23 19-1: Line Segment Intersection Given a set of n line segments, do any of them intersect? What is a brute force method for solving this roblem? How long does it take (if there are n total line segments)

24 19-: Line Segment Intersection Given a set of n line segments, do any of them intersect? What is a brute force method for solving this roblem? Check each air of line segments, see if they intersect using the revious technique How long does it take (if there are n total line segments) Each of the n segments needs to be comarted to n 1 other segments, for a total time of O(n ) We can do better!

25 19-3: Line Segment Intersection Basic idea: Assume that there are no vertical line segments Swee a vertical line across the segments Segment A is above segment B at the line, and as we move the line to the right, Segment B becomes above Segment A, then the segments have crossed

26 19-4: Line Segment Intersection Segment A Segment B Segment B is above Segment A

27 19-5: Line Segment Intersection Segment A Segment B Segment A is above Segment B The two segments must have crossed

28 19-6: Line Segment Intersection Maintain an ordered list of the segments that intersect with the current swee line Whenever two segments become adjacent on this list, check to see if they intersect Only need to check endoints of segments

29 19-7: Line Segment Intersection Maintian a data structure that lets us: Insert a segment s into T Delete a segment s from T Find the segment above s in T Find the segment below s in T Use a red-black tree, using cross roducts to see if segment 1 is above segment at a certain oint

30 19-8: Line Segment Intersection Sort endoints of segments from left to right Break ties: Left endoints before right endoints lowest y-coordinate first for each oint in endoint list if is the left endoint of a segment s Insert s into T if there is a segment above s in T that intersects s or a segment below s in T that intersects s return true if is the right endoint of a segment s if there is a segment above s and below s in T and these segments intersect return true Delete s from T return false

31 19-9: Convex Hull a b d c a a b a b c a b d c

32 19-30: Convex Hull a b c e d a a b a b c a b c d a b e c d a e c d a e d a d

33 19-31: Convex Hull Given a set of oints, what is the smallest convex olygon that contains all oints Alternately, if all of the oints were nails in a board, and we laced a rubber band around all of them, what shae would it form?

34 19-3: Convex Hull

35 19-33: Convex Hull

36 19-34: Convex Hull Several comutational geometry roblems have finding the convex hull as a subroblem Like many grah algorithms have finding a toological sort as a subroblem For instance: Finding the two furthest oints Must lie on the convex hull

37 19-35: Convex Hull Graham s Scan Algorithm Go through all the oints in order Push oints onto a stack Po off oints that don t form art of the convex hull When we re done, stack contains the oints in the convex hull

38 19-36: Convex Hull Gram-Scan Let 0 be the oint with the minimum y-coordinate Sort the oints by increasing olar angle around 0 Push 0, 1, and on the stack S for i 3 to n do while angle formed by to two oints on S doesn t turn left do Po Push( i ) return S

39 19-37: Graham s Scan

40 19-38: Graham s Scan

41 19-39: Graham s Scan Stack

42 19-40: Graham s Scan Stack

43 19-41: Graham s Scan Stack

44 19-4: Graham s Scan Stack

45 19-43: Graham s Scan Stack

46 19-44: Graham s Scan Stack

47 19-45: Graham s Scan Stack

48 19-46: Graham s Scan Stack

49 19-47: Graham s Scan Stack

50 19-48: Graham s Scan Stack

51 19-49: Graham s Scan Stack

52 19-50: Graham s Scan Stack

53 19-51: Graham s Scan Stack

54 19-5: Graham s Scan Stack

55 19-53: Graham s Scan Stack

56 19-54: Graham s Scan Stack

57 19-55: Graham s Scan Stack

58 19-56: Graham s Scan Stack

59 19-57: Graham s Scan Time required: O(nlgn) to sort oints by olar degree Note that you don t need to calculate the olar degree, just determine if one vector is clockwise or counterclockwise of another can be done with a single cross roduct Each element is added to the stack once, and removed at most once (each taking constant time) for a total time of O(n) Total: O(nlgn)

60 19-58: Convex Hull Different Convex Hull algorithm Idea: Attach a string to the lowest oint Rotate string counterclockwise, unti it hits a oint this oint is in the Convex Hull Kee going until the highest oint is reached Continue around back to initial oint

61 19-59: Jarvis s March

62 19-60: Jarvis s March

63 19-61: Jarvis s March

64 19-6: Jarvis s March

65 19-63: Jarvis s March

66 19-64: Jarvis s March

67 19-65: Jarvis s March

68 19-66: Jarvis s March

69 19-67: Jarvis s March

70 19-68: Jarvis s March

71 19-69: Jarvis s March

72 19-70: Jarvis s March How do we determine which oint wras next? When we re going from lowest to highest oint, the smallest olar angle between revious oint and the next oint When going from highest oint back to lowest oint, smallest olar angle (from negative)

73 19-71: Jarvis s March

74 19-7: Jarvis s March

75 19-73: Jarvis s March

76 19-74: Jarvis s March

77 19-75: Jarvis s March

78 19-76: Jarvis s March

79 19-77: Jarvis s March

80 19-78: Jarvis s March

81 19-79: Jarvis s March

82 19-80: Jarvis s March We don t need to actually comute olar angles We just need to comare them, which can be done with a cross roduct From oint k, comaring angles from i and j (going u) Is k i clockwise of k j? (is ( i k ) ( j k ) ositive)?

83 19-81: Jarvis s March Time for Jarvis s march: For each vertex in the convex hull, we need to look at u to n other vertices to find the next vertex in the convex hull. Total time: O(nh), where h is the number of vertices in the convex hull Is this better or worst than Graham s Scan

84 19-8: Closest Pair of Points We have a large number of oints 1,..., n Want to determine which air of oints i, j is closest together How long would a brute force solution take? Can you think of another way?

85 19-83: Closest Pair of Points Divide & Conquer Divide the list oints in half (by a vertical line) Recursively determine the closest air in each half... and then what?

86 19-84: Closest Pair of Points Divide & Conquer Divide the list oints in half (by a vertical line) Recursively determine the closest air in each half Smallest distance between oints is the minimum of: Smallest distance in left half of oints Smallest distance in right half of oints Smallest distance that crosses from left to right

87 19-85: Closest Pair of Points

88 19-86: Closest Pair of Points

89 19-87: Closest Pair of Points To find smallest distance that crosses from left to right: If we comare all n elements in the left sublist with all n elements in the right sublist, how much time would that take?

90 19-88: Closest Pair of Points To find smallest distance that crosses from left to right: If we comare all n elements in the left sublist with all n elements in the right sublist, how much time would that take? Θ(n ), no better than brute force solution!

91 19-89: Closest Pair of Points To find smallest distance that crosses from left to right: Letδ be the smallest distance in the two sublists Examine only the oints that are within δ of the centerline

92 19-90: Closest Pair of Points

93 19-91: Closest Pair of Points δ δ

94 19-9: Closest Pair of Points δ δ

95 19-93: Closest Pair of Points δ δ

96 19-94: Closest Pair of Points δ δ

97 19-95: Closest Pair of Points How many oints can be in the δ δ rectangle? δ δ

98 19-96: Closest Pair of Points How many oints can be in the δ δ rectangle? δ δ

99 19-97: Closest Pair of Points Create two lists of the oints: One sorted by x-coordiate One sorted by y-coordinate Call Find-Smallest using these two lists Find-Smallest(XList,YList)

100 19-98: Closest Pair of Points FindSmallest(L x,l y ) if L x 3 do brute force search on 3 oints Slit list L x in half Put first 1/ in L XL Put second 1/ in L XR Slit list L Y in half For each oint in L y : If L XL, ut in L YL If L XR, ut in L YR δ FindSmallest(L XL,L YL ) δ Min(δ, FindSmallest(L XR,L YR )) δ FindSmallestAcross(L YR,L YR,δ) return δ

101 19-99: Closest Pair of Points Time: Sorting: O(n lg n) using mergesort Recursive call: T(1) = T() = T(3) = c 1 T(n) = T(n/)+c n Θ(nlgn) by the Master Method Total time: O(n lg n)