Divide, and Conquer. Closest Pair

1. CS 321. Algorithm Analysis & Design
Lecture 6
Divide and Conquer

2. Closest Pair

4. Input: A set of n points given in terms of (x,y) coordinates.

5. Input: A set of n points given in terms of (x,y) coordinates.
Output: The closest pair.

6. Input: A set of n points given in terms of (x) coordinates.
Output: The closest pair.

7. Compute distances
between all pairs.
Approach #0
O(n2)

8. Sort the points.
Sweep through adjacent pairs.
Approach 1
Time to sort + O(n)

9. Divide and Conquer.
Split the point set, use recursive solutions.
Approach 2
Cost of Recursion + Time to merge

10. Input: A set of n points given in terms of (x) coordinates.
Output: The closest pair.

11. Input: A set of n points given in terms of (x) coordinates.
Output: The closest pair.

12. Input: A set of n points given in terms of (x) coordinates.
Output: The closest pair.
Closest Pair on Left - d1
Closest Pair on Right - d2

13. Input: A set of n points given in terms of (x) coordinates.
Output: The closest pair.
Closest Pair on Left - d1
Closest Pair on Right - d2

14. Input: A set of n points given in terms of (x,y) coordinates.
Output: The closest pair.

15. Input: A set of n points given in terms of (x,y) coordinates.
Output: The closest pair.
1
2
3
4
5
6
7
8
9
10
11
12
13

16. Input: A set of n points given in terms of (x,y) coordinates.
Output: The closest pair.

17. Input: A set of n points given in terms of (x,y) coordinates.
Output: The closest pair.
13
1
7
5
3
9
8
11
10
4
12
2
6

18. Input: A set of n points given in terms of (x,y) coordinates.
Output: The closest pair.

19. Input: A set of n points given in terms of (x,y) coordinates.
Output: The closest pair.

20. Input: A set of n points given in terms of (x,y) coordinates.
Output: The closest pair.

21. Input: A set of n points given in terms of (x,y) coordinates.
Output: The closest pair.

22. Input: A set of n points given in terms of (x,y) coordinates.
Output: The closest pair.

23. Input: A set of n points given in terms of (x,y) coordinates.
Output: The closest pair.
d/2
d/2

24. Input: A set of n points given in terms of (x,y) coordinates.
Output: The closest pair.
d/2
d/2

32. T(n) ≤ 2T(n/2) + O(n)
T(n) = O(n log n)