1. Chan’s algorithm
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2. Graham’s Scan : 𝑂(𝑛 log 𝑛 ) Jarvis’s March : 𝑂(𝑛ℎ)
Planar Convex Hull
Graham’s Scan : 𝑂(𝑛 log 𝑛 )
Jarvis’s March : 𝑂(𝑛ℎ)
Is there any 𝑂(𝑛 log ℎ ) algorithm?
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3. 3 stages of Chan’s algorithm
combining Graham’s scan and Jarvis’s March together
𝑂 𝑛 log ℎ
3 stages of Chan’s algorithm
divide vertices into partitions
apply Graham’s scan on each partition
apply Jarvis’s March on the small convex hull
(repeat 1~3 until we find the hull)
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4. Consider arbitrary value 𝑚<𝑛, the size of partition
Chan’s Algorithm
Stage1 : Partition
Consider arbitrary value 𝑚<𝑛, the size of partition
how to decide 𝑚 will be treated later
Partition the points into groups, each of size 𝑚
𝑟= 𝑛 𝑚 is the number of groups
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5. Chan’s Algorithm
Stage 1
n = 32
Set m = 8
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6. Chan’s Algorithm
Stage 1
n = 32
Set m = 8
r = 4
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7. Compute convex hull of each partition using Graham’s scan
Chan’s Algorithm
Stage2 : Graham’s Scan
Compute convex hull of each partition using Graham’s scan
Total 𝑂(𝑛 log 𝑚 ) time
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8. Chan’s Algorithm Stage 2 (After Stage 1) m = 8 r = 4
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9. Chan’s Algorithm Stage 2 Using Graham’s Scan O 𝑚 log 𝑚 for each group
-> total 𝑂(𝑚𝑟 log 𝑚 )
= 𝑂(𝑛 log 𝑚 )
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10. How to merge these r hulls into a single hull?
Chan’s Algorithm
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)
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11. Chan’s Algorithm (-inf,0) -> lowest pt (−∞,0) lowest pt
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12. Find the point that maximize the angle in each hull
Chan’s Algorithm
Find the point that maximize the angle in each hull
(−∞,0) 1
lowest pt
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13. Find the point that maximize the angle in each hull
Chan’s Algorithm
Find the point that maximize the angle in each hull
(−∞,0) 2 1
lowest pt
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14. Find the point that maximize the angle in each hull
Chan’s Algorithm
Find the point that maximize the angle in each hull 3
(−∞,0) 2 1
lowest pt
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15. Chan’s Algorithm If 𝑚<ℎ, then the algorithm will fail!
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16. FAIL EXAMPLE – too small value m
Chan’s Algorithm
FAIL EXAMPLE – too small value m
m = 4
(−∞,0)
4 iteration
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17. Chan’s Algorithm
In 4(a), how to find such points?
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18. Find the point that maximize the angle in each hull
Chan’s Algorithm
Find the point that maximize the angle in each hull
(−∞,0) 1
lowest pt
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19. Find the point that maximize the angle in a hull
Chan’s Algorithm
Find the point that maximize the angle in a hull
(−∞,0)
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20. Finding tangent between a point and a convex 𝑚-gon
Chan’s Algorithm
Finding tangent between a point and a convex 𝑚-gon 5 1
𝑂 log 𝑚 process 4 3 2
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21. Chan’s Algorithm
→ 𝑂( log 𝑚 )
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22. Chan’s Algorithm Analysis Suppose God told you some good value for 𝑚
ℎ≤𝑚≤ ℎ 2
𝑂 𝑛 log 𝑚 for stage1~2
At most h steps in Jarvis’s March
𝑂( log 𝑚 ) time to compute each tangent
𝑟 tangent for each iteration
total 𝑂 ℎ𝑟 log 𝑚 =𝑂(ℎ 𝑛 𝑚 log 𝑚 ) time
𝑂 𝑛+ℎ 𝑛 𝑚 log 𝑚 =𝑂 𝑛 log 𝑚 =𝑂(𝑛 log ℎ ) as desired.
𝑚≤ ℎ 2
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23. The only question remaining is…
Chan’s Algorithm
The only question remaining is…
how do we know what value to give to 𝑚?
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24. Then we eventually get 𝑚 to be in [ℎ, ℎ 2 ] !
Chan’s Algorithm
Answer : square search
Try this way - 𝑚=2, 4, 8, …, 𝑡 , …
Then we eventually get 𝑚 to be in [ℎ, ℎ 2 ] !
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25. The algorithm stops as soon as 2 2 𝑡 ≥ℎ
Chan’s Algorithm
The algorithm stops as soon as 𝑡 ≥ℎ
𝑡= lg lg ℎ
Total time complexity
𝑡=1 lg lg ℎ 𝑛 log 𝑡 = 𝑡=1 lg lg ℎ 𝑛 2 𝑡 ≤𝑛 lg lg ℎ =2𝑛 2 lg lg ℎ =2𝑛 log ℎ =𝑂(𝑛 log ℎ )
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