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Chanโs algorithm CS504 Presentation

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**Grahamโs Scan : ๐(๐ log ๐ ) Jarvisโs March : ๐(๐โ)**

Planar Convex Hull Grahamโs Scan : ๐(๐ log ๐ ) Jarvisโs March : ๐(๐โ) Is there any ๐(๐ log โ ) algorithm? CS504 Presentation

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**3 stages of Chanโs algorithm**

combining Grahamโs scan and Jarvisโs March together ๐ ๐ log โ 3 stages of Chanโs algorithm divide vertices into partitions apply Grahamโs scan on each partition apply Jarvisโs March on the small convex hull (repeat 1~3 until we find the hull) CS504 Presentation

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**Consider arbitrary value ๐<๐, the size of partition**

Chanโs Algorithm Stage1 : Partition Consider arbitrary value ๐<๐, the size of partition how to decide ๐ will be treated later Partition the points into groups, each of size ๐ ๐= ๐ ๐ is the number of groups CS504 Presentation

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Chanโs Algorithm Stage 1 n = 32 Set m = 8 CS504 Presentation

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Chanโs Algorithm Stage 1 n = 32 Set m = 8 r = 4 CS504 Presentation

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**Compute convex hull of each partition using Grahamโs scan **

Chanโs Algorithm Stage2 : Grahamโs Scan Compute convex hull of each partition using Grahamโs scan Total ๐(๐ log ๐ ) time CS504 Presentation

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**Chanโs Algorithm Stage 2 (After Stage 1) m = 8 r = 4**

CS504 Presentation

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**Chanโs Algorithm Stage 2 Using Grahamโs Scan O ๐ log ๐ for each group**

-> total ๐(๐๐ log ๐ ) = ๐(๐ log ๐ ) CS504 Presentation

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

Chanโs Algorithm Stage3 : Jarvisโs March How to merge these r hulls into a single hull? IDEA : treat each hull as a โfat pointโ and run Jarvisโs March! # of iteration is at most m to guarantee the time complexity O(nlogh) CS504 Presentation

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**Chanโs Algorithm (-inf,0) -> lowest pt (โโ,0) lowest pt**

CS504 Presentation

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**Find the point that maximize the angle in each hull**

Chanโs Algorithm Find the point that maximize the angle in each hull (โโ,0) 1 lowest pt CS504 Presentation

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**Find the point that maximize the angle in each hull**

Chanโs Algorithm Find the point that maximize the angle in each hull (โโ,0) 2 1 lowest pt CS504 Presentation

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**Find the point that maximize the angle in each hull**

Chanโs Algorithm Find the point that maximize the angle in each hull 3 (โโ,0) 2 1 lowest pt CS504 Presentation

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**Chanโs Algorithm If ๐<โ, then the algorithm will fail!**

CS504 Presentation

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**FAIL EXAMPLE โ too small value m**

Chanโs Algorithm FAIL EXAMPLE โ too small value m m = 4 (โโ,0) 4 iteration CS504 Presentation

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Chanโs Algorithm In 4(a), how to find such points? CS504 Presentation

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**Find the point that maximize the angle in each hull**

Chanโs Algorithm Find the point that maximize the angle in each hull (โโ,0) 1 lowest pt CS504 Presentation

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**Find the point that maximize the angle in a hull**

Chanโs Algorithm Find the point that maximize the angle in a hull (โโ,0) CS504 Presentation

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**Finding tangent between a point and a convex ๐-gon**

Chanโs Algorithm Finding tangent between a point and a convex ๐-gon 5 1 ๐ log ๐ process 4 3 2 CS504 Presentation

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Chanโs Algorithm โ ๐( log ๐ ) CS504 Presentation

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**Chanโs Algorithm Analysis Suppose God told you some good value for ๐**

โโค๐โค โ 2 ๐ ๐ log ๐ for stage1~2 At most h steps in Jarvisโs March ๐( log ๐ ) time to compute each tangent ๐ tangent for each iteration total ๐ โ๐ log ๐ =๐(โ ๐ ๐ log ๐ ) time ๐ ๐+โ ๐ ๐ log ๐ =๐ ๐ log ๐ =๐(๐ log โ ) as desired. ๐โค โ 2 CS504 Presentation

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**The only question remaining isโฆ **

Chanโs Algorithm The only question remaining isโฆ how do we know what value to give to ๐? CS504 Presentation

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**Then we eventually get ๐ to be in [โ, โ 2 ] !**

Chanโs Algorithm Answer : square search Try this way - ๐=2, 4, 8, โฆ, ๐ก , โฆ Then we eventually get ๐ to be in [โ, โ 2 ] ! CS504 Presentation

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**The algorithm stops as soon as 2 2 ๐ก โฅโ**

Chanโs Algorithm The algorithm stops as soon as ๐ก โฅโ ๐ก= lg lg โ Total time complexity ๐ก=1 lg lg โ ๐ log ๐ก = ๐ก=1 lg lg โ ๐ 2 ๐ก โค๐ lg lg โ =2๐ 2 lg lg โ =2๐ log โ =๐(๐ log โ ) CS504 Presentation

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