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

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

If p1 = (x1, y1) and p2 = (x2, y2) then p1  p2 = x1 x2 = x1y2 – x2y1* y1 y2 If p1  p2 > 0, then p1 is clockwise from p2 (turning around the origin); if p1  p2 < 0, then p1 is counterclockwise from p2; If p1  p2 = 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 |x1y2 – x2y1|. Here we simplify for 2D.

If DIRECTION(pi, pj, pk) > 0, then pj – pi is a left turn from pk – pi around pi < 0, then pj – pi is a right turn from pk – pi around pi = 0, then pj – pi and pk – pi are collinear

How?!

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.

Jarvis’s march is output-sensitive!

Closest Pair by Divide-and-Conquer Following slides are copied from Dariusz Kowalski (www.csc.liv.ac.uk/~darek/COMP523/lecture5.ppt)

Lecture 5: Divide and Conquer, cont. (Dariusz Kowalski) 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 Lecture 5: Divide and Conquer, cont. (Dariusz Kowalski)

Lecture 5: Divide and Conquer, cont. (Dariusz Kowalski) 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)

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    Lecture 5: Divide and Conquer, cont. (Dariusz Kowalski) L

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 /2 /2 /2 /2 Lecture 5: Divide and Conquer, cont. (Dariusz Kowalski) L

Problems Ex. 33.3-2 Ex. 33.3-4 Ex. 33.3-5 Ex. 33.3-6 Ex. 33.4-3