Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into...

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1/12/16 CMPS 6640/4040: Computational Geometry 1 CMPS 6640/4040: Computational Geometry Spring 2016 Convex Hulls Carola Wenk

Transcript of Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into...

Page 1: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 1

CMPS 6640/4040: Computational GeometrySpring 2016

Convex HullsCarola Wenk

Page 2: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 2

Convex Hull Problem

Given a set of pins on a pinboard

and a rubber band around them.

How does the rubber band look when it snaps tight?

The convex hull of a point set is one of the simplest shape approximations for a set of points.

Page 3: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 3

Convexity

A set C R2 is convex if for every two points p,qC the line segment pq is fully contained in C.

convex non-convex

Page 4: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 4

Convex Hull

The convex hull CH(P) of a point set P R2 is the smallest convex set C P. In other words CH(P) = C .

C PC convex

P

Page 5: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 5

Convex Hull Observation: CH(P) is the unique convex polygon whose vertices are points of P and which contains all points of P.

0

2

1

34

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Goal: Compute CH(P). What does that mean? How do we represent/store CH(P)?

Represent the convex hull as the sequence of points on the convex hull polygon (the boundary of the convex hull), in counter-clockwise order.

Page 6: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 6

Orientation Test / Halfplane Test

p

q

r

r

q

p

• positive orientation(counter-clockwise)

• r lies to the left of pq

• negative orientation(clockwise)

• r lies to the right of pq

rq

p• zero orientation• r lies on the line pq

• Orient(p,q,r) = sign det

• Can be computed in constant time

1 px py1 qx qy1 rx ry

,where p = (px,py)

Page 7: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 7

Graham’s Scan

Another incremental algorithm– Compute solution by incrementally adding points – Add points in which order?

• Sorted by x-coordinate• But convex hulls are cyclically ordered Split convex hull into upper and lower part

upper convex hull UCH(P)

lower convex hull LCH(P)

Page 8: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 8

Graham’s LCHAlgorithm Grahams_LCH(P):// Incrementally compute the lower convex hull of P Input: Point set P R2

Output: A stack S of vertices describing LCH(P) in counter-clockwise order

Sort P in increasing order by x-coordinate P = {p1,…,pn}S.push(p1)S.push(p2)for i=3 to n

while |S|>=2 and orientation(S.second(), S.top(), pi,) <= 0 // no left turnS.pop()

S.push(pi)

• Each element is appended only once, and hence only deleted at most once the for-loop takes O(n) time

• O(n log n) time total

O(n log n)

O(n)

Page 9: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 9

Graham’s Scanp7

p1

p2

p3

p4

p5

p6

p8

p9

p10

p11

p12

p1

p2

No left turn

p7

p1

p2

p3

p4

p5

p6

p8

p9

p10

p11

p12

p1

p7

p1

p2

p3

p4

p5

p6

p8

p9

p10

p11

p12

p1

Left turn

pop push

p3push

p7

p1

p2

p3

p4

p5

p6

p8

p9

p10

p11

p12

p1

p3

p4

No left turn

pop

Page 10: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 10

Graham’s Scanp7

p1

p2

p3

p4

p5

p6

p8

p9

p10

p11

p12

p1

p3

No left turn

pop

p7

p1

p2

p3

p4

p5

p6

p8

p9

p10

p11

p12

p1

push

p7

p1

p2

p3

p4

p5

p6

p8

p9

p10

p11

p12

p1

Left turn

pushp5

p7

p1

p2

p3

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p5

p6

p8

p9

p10

p11

p12

p1

Left turn

pushp1

p5p5

p6p6

p7

Page 11: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 11

Graham’s Scanp7

p1

p2

p3

p4

p5

p6

p8

p9

p10

p11

p12

No left turn

p1

p5

p6

p7 popp1

p2

p3

p4

p5

p6

p8

p9

p10

p11

p12

Left turn

p1

p5

p6 push

p1

p2

p3

p4

p5

p6

p8

p9

p10

p11

p12

No left turn

p1

p5

p6

p8

popp1

p2

p3

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p5

p6

p8

p9

p10

p11

p12

No left turn

p1

p5

p6 pop

Page 12: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 12

Graham’s Scan

p1

p2

p3

p4

p5

p6

p8

p9

p10

p11

p12

Left turn

p1

p5

pushp1

p2

p3

p4

p5

p6

p8

p9

p10

p11

p12

Left turn

p1

p5

pushp9

p1

p2

p3

p4

p5

p6

p8

p9

p10

p11

p12

No left turn

p1

p5

popp9

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p1

p2

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p10

p11

p12

No left turn

p1

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popp9

Page 13: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 13

Graham’s Scan

p1

p2

p3

p4

p5

p6

p8

p9

p10

p11

p12

Left turn

p1

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push

p1

p2

p3

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p5

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Left turn

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pushp11

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Page 14: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 14

Convex Hull: Divide & Conquer Preprocessing: sort the points by x-coordinate

Divide the set of points into two sets A and B:

A contains the left n/2 points,

B contains the right n/2 points

Recursively compute the convex hull of A

Recursively compute the convex hull of B

Merge the two convex hulls

A B

Page 15: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 15

Merging Find upper and lower tangent

With those tangents the convex hull of AB can be computed from the convex hulls of A and the convex hull of B in O(n) linear time

A B

Page 16: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 16

check with orientation test

right turnleft turn

Finding the lower tangent a = rightmost point of Ab = leftmost point of Bwhile T=ab not lower tangent to both

convex hulls of A and B do{while T not lower tangent to convex hull of A do{

a=a-1}while T not lower tangent to convex hull of B do{b=b+1

}}

A B0

a=2

1

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3

4

0

1

2

3

4=b

5

67

Page 17: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 17

Convex Hull: Runtime Preprocessing: sort the points by x-coordinate

Divide the set of points into two sets A and B:

A contains the left n/2 points,

B contains the right n/2 points

Recursively compute the convex hull of A

Recursively compute the convex hull of B

Merge the two convex hulls

O(n log n) just once

O(1)

T(n/2)

T(n/2)

O(n)

Page 18: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 18

Convex Hull: Runtime Runtime Recurrence:

T(n) = 2 T(n/2) + cn

Solves to T(n) = (n log n)

Page 19: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 19

Master theoremT(n) = a T(n/b) + f (n)

CASE 1: f (n) = O(nlogba – ) T(n) = (nlogba) .

CASE 2: f (n) = (nlogba logkn) T(n) = (nlogba logk+1n) .

CASE 3: f (n) = (nlogba + ) and a f (n/b) c f (n) T(n) = ( f (n)) .

, where a 1, b > 1, and f is asymptotically positive.

Convex hull: a = 2, b = 2 nlogba = n CASE 2 (k = 0) T(n) = (n log n) .

Page 20: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 20

Jarvis’ March (Gift Wrapping)Algorithm Giftwrapping_CH(P):// Compute CH(P) by incrementally inserting points from left to right Input: Point set P R2

Output: List q1, q2,… of vertices in counter-clockwise order around CH(P)q1 = point in P with smallest y (if ties, with smallest x)q2 = point in P with smallest angle to horizontal line through q1i = 2do {

i++qi = point with smallest angle to line through qi-2 and qi-1

} while qi ≠ q1

q1

q2

q3

• Runtime: O(hn) , where n = |P| and h = #points on CH(P)• Output-sensitive algorithm

Page 21: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 21

Chan’s Algorithm• Runtime goal: O(n log h) , where n = |P| and h = #points on CH(P)• Output-sensitive algorithm

Page 22: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 22

Lower Bound• Comparison-based sorting of n elements takes

(n log n) time. • How can we use this lower bound to show a lower

bound for the computation of the convex hull of npoints in R2?

Page 23: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 23

Lower Bound• Comparison-based sorting of n elements takes

(n log n) time. • How can we use this lower bound to show a lower

bound for the computation of the convex hull of npoints in R2?

• Devise a sorting algorithm which uses the convex hull and otherwise only linear-time operations Since this is a comparison-based sorting algorithm, the

lower bound (n log n) applies Since all other operations need linear time, the convex

hull algorithm has to take (n log n) time

Page 24: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 24

CH_SortAlgorithm CH_Sort(S):/* Sorts a set of numbers using a convex hull

algorithm. Converts numbers to points, runs CH, converts back to sorted sequence. */

Input: Set of numbers S ROutput: A list L of of numbers in S sorted in

increasing orderP=for each sS insert (s,s2) into PL’ = CH(P) // compute convex hullFind point p’P with minimum x-coordinatefor each p=(px,py)L’, starting with p’,

add px into Lreturn L

s2

s-2-4 1 4 5

Page 25: Convex Hulls - Tulane University• But convex hulls are cyclically ordered Split convex hull into upper and lower part upper convex hull UCH(P) lower convex hull LCH(P) 1/12/16 CMPS

1/12/16 CMPS 6640/4040: Computational Geometry 25

Convex Hull Summary• Graham’s scan: O(n log n)• Divide-and-conquer: O(n log n) • Jarvis’ march (gift wrapping): O(nh)• Chan’s algorithm: O(n log h)

• Lower bound: (n log n)


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