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Chan’s algorithm

Planar Convex Hull

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• Graham’s Scan :

• Jarvis’s March :

• Is there any algorithm?

Chan’s Algorithm

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• Chan’s algorithm– combining Graham’s scan and Jarvis’s March together

• 3 stages of Chan’s algorithm1. divide vertices into partitions2. apply Graham’s scan on each partition3. apply Jarvis’s March on the small convex hull

(repeat 1~3 until we find the hull)

Chan’s Algorithm

CS504 Presentation

• Stage1 : Partition

• Consider arbitrary value , the size of partition– how to decide will be treated later

• Partition the points into groups, each of size – is the number of groups

Chan’s Algorithm

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• Stage 1

n = 32

Set m = 8

Chan’s Algorithm

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• Stage 1

n = 32

Set m = 8 r = 4

Chan’s Algorithm

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• Stage2 : Graham’s Scan

• Compute convex hull of each partition using Gra-ham’s scan

• Total time

Chan’s Algorithm

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• Stage 2

(After Stage 1) m = 8 r = 4

Chan’s Algorithm

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• Stage 2

Using Graham’s Scan

for each group-> total =

Chan’s Algorithm

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• Stage3 : Jarvis’s March

• How to merge these r hulls into a single hull?

• IDEA : treat each hull as a “fat point” and run Jarvis’s March!

• # of iteration is at most m– to guarantee the time complexity O(nlogh)

Chan’s Algorithm

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• (-inf,0) -> lowest pt

(−∞ ,0)

lowest pt

Chan’s Algorithm

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• Find the point that maximize the angle in each hull

(−∞ ,0)

lowest pt1

Chan’s Algorithm

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• Find the point that maximize the angle in each hull

(−∞ ,0)

lowest pt1

2

Chan’s Algorithm

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• Find the point that maximize the angle in each hull

(−∞ ,0)

lowest pt1

2

3

Chan’s Algorithm

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If , then the algorithm will fail!

Chan’s Algorithm

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• FAIL EXAMPLE – too small value m

m = 4

(−∞ ,0)

4 iteration

Chan’s Algorithm

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In 4(a), how to find such points?

Chan’s Algorithm

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• Find the point that maximize the angle in each hull

(−∞ ,0)

lowest pt1

Chan’s Algorithm

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• Find the point that maximize the angle in a hull

(−∞ ,0)

Chan’s Algorithm

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• Finding tangent between a point and a convex -gon

1

23

4

5

process

Chan’s Algorithm

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→𝑂 (log𝑚)

Chan’s Algorithm

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Analysis• Suppose God told you some good value for

• for stage1~2• At most h steps in Jarvis’s March

– time to compute each tangent– tangent for each iteration– total time

• as desired.

𝑚≤h2

Chan’s Algorithm

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• The only question remaining is…

how do we know what value to give to ?

Chan’s Algorithm

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• Answer : square search

• Try this way -

• Then we eventually get to be in !

Chan’s Algorithm

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• The algorithm stops as soon as

• Total time complexity