Your SlideShare is downloading. ×
Permuting Polygons
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×

Introducing the official SlideShare app

Stunning, full-screen experience for iPhone and Android

Text the download link to your phone

Standard text messaging rates apply

Permuting Polygons

486
views

Published on

On a problem of polygons, convexity, and computational complexity.

On a problem of polygons, convexity, and computational complexity.


0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
486
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
11
Comments
0
Likes
0
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 1. Permuting Polygons Thomas Henderson Under the direction of Dr. Paul Latiolais Second reader Dr. Bin Jiang
  • 2. types of polygons
  • 3. a polygon is simple if it does not self-intersect. simple
  • 4. a polygon is simple if it does not self-intersect. simple not simple
  • 5. a polygon is simple if it does not self-intersect. simple really not simple not simple
  • 6. a polygon is convex if, given two points in the polygon, the line segment joining them is also in the polygon.
  • 7. a polygon is convex if, given two points in the polygon, the line segment joining them is also in the polygon. convex
  • 8. a polygon is convex if, given two points in the polygon, the line segment joining them is also in the polygon. convex
  • 9. a polygon is convex if, given two points in the polygon, the line segment joining them is also in the polygon. convex
  • 10. a polygon is convex if, given two points in the polygon, the line segment joining them is also in the polygon. not convex convex
  • 11. a polygon is star-shaped if all points in the polygon can be seen from some point in the polygon's interior.
  • 12. a polygon is star-shaped if all points in the polygon can be seen from some point in the polygon's interior. k
  • 13. a polygon is star-shaped if all points in the polygon can be seen from some point in the polygon's interior. k
  • 14. a polygon is star-shaped if all points in the polygon can be seen from some point in the polygon's interior. k k is in the kernel of the polygon.
  • 15. this polygon is NOT star-shaped.
  • 16. this polygon is NOT star-shaped.
  • 17. this polygon is NOT star-shaped.
  • 18. this polygon is NOT star-shaped. the kernel is empty.
  • 19. a polygon can be oriented by adding a direction to every edge.
  • 20. a polygon can be oriented by adding a consistent direction to every edge.
  • 21. a polygon can be oriented by adding a consistent direction to every edge. the polygon is oriented (clockwise).
  • 22. edge swaps
  • 23. let P be a clockwise-oriented, star-shaped polygon. let a and b be edges of P which are adjacent, and which form a left-hand turn. let k be a point in the kernel of P.
  • 24. • the new polygon contains the old one
  • 25. • the new polygon contains the old one
  • 26. • the new polygon contains the old one
  • 27. • the new polygon contains the old one
  • 28. • the new polygon contains the old one • the new kernel contains the old one
  • 29. • the new polygon contains the old one • the new kernel contains the old one • the new polygon is star- shaped
  • 30. convexification
  • 31. Problem: Given a star- shaped polygon, can you make it a convex polygon by swapping edges?
  • 32. Problem: Given a star- shaped polygon, can you make it a convex polygon by swapping edges? no, seriously: can you?
  • 33. Instructions: • Make a star-shaped polygon. • Turn it into a convex polygon. You may ONLY swap adjacent edges!
  • 34. Instructions: • Make a star-shaped polygon. • Turn it into a convex polygon. You may ONLY swap adjacent edges! go!
  • 35. The Convexification Algorithm
  • 36. The Convexification Algorithm Traverse the polygon in the direction it is oriented. When you come to a turn: • if the turn is a RHT, do nothing and continue • if the turn is a LHT, swap the edges and continue
  • 37. Theorem: The Convexification Algorithm will convexify any star-shaped polygon.
  • 38. The Idea of the Proof: Show that any two edges of any star-shaped polygon will be swapped at most once.
  • 39. let P be a clockwise-oriented, star-shaped polygon. let a and b be edges of P which are adjacent. let k be a point in the kernel of P. let L be a line through k, and parallel to a.
  • 40. Case 1: a and b form a RHT
  • 41. Case 1: a and b form a RHT
  • 42. Case 1: a and b form a RHT ZERO SWAPS
  • 43. Case 2: a and b form a LHT
  • 44. Case 2: a and b form a LHT
  • 45. Case 2: a and b form a LHT
  • 46. ONE SWAP
  • 47. ? ?? ? ? ? ?
  • 48. ? ?? ? ? ? ? impossible!
  • 49. if a encounters any RHTs along the way, it stops.
  • 50. if a encounters any RHTs along the way, it stops. what if there are ONLY LHTs?
  • 51. contradiction!
  • 52. contradiction! (the polygon was assumed to be star- shaped)
  • 53. analysis of algorithms
  • 54. What is the worst possible behavior of the Convexification Algorithm?
  • 55. What is the worst possible behavior of the Convexification Algorithm?
  • 56. Suppose P has n sides. If the algorithm must swap every side with every other side, the number of swaps is (n - 1) + (n - 2) + ... + 2 + 1 = n(n - 1)/2 = n2 /2 - n/2
  • 57. Suppose P has n sides. If the algorithm must swap every side with every other side, the number of swaps is (n - 1) + (n - 2) + ... + 2 + 1 = n(n - 1)/2 2 = n2 /2 - n/2 O(n )
  • 58. 2
  • 59. (n - 3) + (n - 4) + ... + 2 + 1 = (n - 3)(n - 2)/2 = 1/2n2 - 5/2n + 3 2 O(n )