1. Data Structures and Algorithms (AT70.02) Comp. Sc. and Inf. Mgmt. Asian Institute of Technology Instructor: Dr. Sumanta Guha Slide Sources: CLRS “Intro. To Algorithms” book website (copyright McGraw Hill) adapted and supplemented

2. CLRS “Intro. To Algorithms” Ch. 33: Computational Geometry

3. If p 1 = (x 1, y 1 ) and p 2 = (x 2, y 2 ) then p 1  p 2 = x 1 x 2 = x 1 y 2 – x 2 y 1 * y 1 y 2 If p 1  p 2 > 0, then p1 is clockwise from p2 (turning around the origin); if p 1  p 2 < 0, then p1 is counterclockwise from p2; If p 1  p 2 = 0, then p1 and p2 are collinear (pointing in either same or opposite directions) * The cross product is actually a 3D vector whose magnitude is |x 1 y 2 – x 2 y 1 |. Here we simplify for 2D.

5. If DIRECTION(p i, p j, p k ) > 0, then p j – p i is a left turn from p k – p i around p i < 0, then p j – p i is a right turn from p k – p i around p i = 0, then p j – p i and p k – p i are collinear

7. How?!

10. Eliminate the polar co-ordinate sort part of Graham’s scan by modifying it to scan the vertices sorted from left to right. Otherwise, the underlying principle, data structures, etc. remain the same. Comment : This modification of Graham’s scan was made by Andrew (1979) and is often called the Monotone Chain algorithm.

11. Jarvis’s march is output-sensitive!

12. Closest Pair by Divide-and-Conquer Following slides are copied from Dariusz Kowalski (www.csc.liv.ac.uk/~darek/COMP523/le cture5.ppt)

13. Lecture 5: Divide and Conquer, cont. (Dariusz Kowalski) 13 Solution Preprocessing: Sort points according to the first coordinate (obtain horizontal list H) and according the second coordinate (obtain vertical list V) Main Algorithm: Partition list H input into halves in linear time (draw vertical line L containing medium point) Solve the problem for each half of the input separately, obtaining two pairs of closest points; let  be the smallest distance from the obtained ones Find if there is a pair which has distance smaller than  - if yes then find the smallest distance pair

14. 14 Main difficulty Find if there is a pair which has distance smaller than  - if yes then find the smallest distance pair How to do it in linear time?   Lecture 5: Divide and Conquer, cont. (Dariusz Kowalski)

15. 15 Closest pair in the strip 1. Find a sublist P of V containing all points with a distance at most  from the line L - in linear time 2. For each element in P check its distance to the next 8 elements and remember the closest pair ever found   L Lecture 5: Divide and Conquer, cont. (Dariusz Kowalski)

16. 16 Why to check only 8 points ahead? 1. Partition the strip into squares of length  /2 as shown in the picture. 2. Each square contains at most 1 point - by definition of . 3. If there are at least 2 squares between points then they can not be the closest points. 4. There are at most 8 squares to check.  /2/2 L /2/2 /2/2 /2/2 /2/2 /2/2 Lecture 5: Divide and Conquer, cont. (Dariusz Kowalski)

17. Problems Ex. 33.3-2 Ex. 33.3-4 Ex. 33.3-5 Ex. 33.3-6 Ex. 33.4-3 Ex. 33.4-4 Ex. 33.4-5 Prob. 33-1