Download presentation

Presentation is loading. Please wait.

1
Chan’s algorithm CS504 Presentation

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? CS504 Presentation

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) CS504 Presentation

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 CS504 Presentation

5
Chan’s Algorithm Stage 1 n = 32 Set m = 8 CS504 Presentation

6
Chan’s Algorithm Stage 1 n = 32 Set m = 8 r = 4 CS504 Presentation

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 CS504 Presentation

8
**Chan’s Algorithm Stage 2 (After Stage 1) m = 8 r = 4**

CS504 Presentation

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 ) CS504 Presentation

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) CS504 Presentation

11
**Chan’s Algorithm (-inf,0) -> lowest pt (−∞,0) lowest pt**

CS504 Presentation

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 CS504 Presentation

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 CS504 Presentation

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 CS504 Presentation

15
**Chan’s Algorithm If 𝑚<ℎ, then the algorithm will fail!**

CS504 Presentation

16
**FAIL EXAMPLE – too small value m**

Chan’s Algorithm FAIL EXAMPLE – too small value m m = 4 (−∞,0) 4 iteration CS504 Presentation

17
Chan’s Algorithm In 4(a), how to find such points? CS504 Presentation

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 CS504 Presentation

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) CS504 Presentation

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 CS504 Presentation

21
Chan’s Algorithm → 𝑂( log 𝑚 ) CS504 Presentation

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 CS504 Presentation

23
**The only question remaining is… **

Chan’s Algorithm The only question remaining is… how do we know what value to give to 𝑚? CS504 Presentation

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 ] ! CS504 Presentation

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 ) CS504 Presentation

Similar presentations

OK

1 Convex Hull in Two Dimensions Jyun-Ming Chen Refs: deBerg et al. (Chap. 1) O’Rourke (Chap. 3)

1 Convex Hull in Two Dimensions Jyun-Ming Chen Refs: deBerg et al. (Chap. 1) O’Rourke (Chap. 3)

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on standing order Ppt on print culture and the modern world download Ppt on file system in linux Ppt on project management process Ppt on world book day usa Ppt on pierre simon laplace Download ppt on indian mathematicians and their contributions Ppt on panel discussion moderator Ppt on steve jobs biography page Ppt on unity in diversity and organic farming