1. Lecture 4 Divide-and-Conquer

2. What’s Self-Reducibility?
A problem can be reduced to several “same” problems with smaller inputs.

3. Divide and Conquer Divide the problem into subproblems.
Conquer the subproblems by solving them recursively.
Combine the solutions to subproblems into the solution for original problem.

4. Tree Structure problem subproblem subproblem subsubproblem

5. Insertion Sort

6. key

7. Merge Sort

8. Procedure
divide
conquer
combine

9. Heapsort 16 14 10 3 8 7 9 2 4 1

10. Heapsort

11. Heapsort

12. Quicksort
divide
conquer
combine is simplified

13. Selection with O(n) Comparisons
divide
conquer
divide
conquer

14. Remark on Divide and Conquer
Key Point:

15. Merge Sort
Quick Sort
Selection
Analysis requires to solve a recurrence!

16. Maximum Subarray

17. Problem

20. Closest pair of points

24. Matrix Multiplication

25. n/2
Step 1
n/2

26. Step 2
Step 3

27. Step 4

28. Master Theorem