1. Chapter 4: Divide and Conquer

2. Divide and Conquer

3. Divide and Conquer
(Normally, they can be ignored.)

4. Divide and Conquer Methods for solving recurrences:
Substitution Method: Guess a bound and use Mathematical induction to prove.
Master Method: Memorize three cases and use them to solve recurrences of the form: T(n) = a T(n/b) + f(n)
Iteration Method: However, it is too easy to make an error in parenthesization, and that recursion trees give a better intuitive idea than iterating the recurrence of how the recurrence progresses.
Recursion-tree Method: Converts the recurrence into a tree whose nodes represent the cost incurred at various levels of recursion. We use techniques for bounding summations to solve the recurrence.

5. The Maximum-subarray problem
Consider investing in stock market. You can buy one unit of stock only one time and then sell it at a later date, buying and selling after close of the trading day. Suppose you have a “Cristal Ball” to see the price of the stock in the future. What is your strategy?

6. The Maximum-subarray problem
Should you always buy at the lowest or sell at the highest? Consider:
We need to find the nonempty, contiguous subarray of array A whose values have the largest sum.

7. The Maximum-subarray problem
Three possibilities for the location of the maximum-subarray with respect to the midpoint (the divide point):

8. The Maximum-subarray problem

9. The Maximum-subarray problem

10. The Maximum-subarray problem

11. The Maximum-subarray problem

12. The Maximum-subarray problem

13. The Maximum-subarray problem

14. Strassen’s algorithm for matrix multiplication

15. Strassen’s algorithm for matrix multiplication

16. Strassen’s algorithm for matrix multiplication

17. Strassen’s algorithm for matrix multiplication
Note: We can partition matrices without copying entries by instead using
index calculations. It would take only constant time, instead of Θ( 𝑛 2 ) time. However, the asymptotic analysis won’t change when we use either technique.

18. Strassen’s algorithm for matrix multiplication
n x n

19. Strassen’s algorithm for matrix multiplication

20. Strassen’s algorithm for matrix multiplication

21. Strassen’s algorithm for matrix multiplication

22. Strassen’s algorithm for matrix multiplication

23. Strassen’s algorithm for matrix multiplication
To see how these computations work, expand each right-hand side, replacing
each Pi with the sub-matrices of A and B that form it, and cancel terms:

24. Strassen’s algorithm for matrix multiplication

25. Substitution method 1. Guess the solution.
2. Use induction to find the constants and show that the solution works.

26. Substitution method

27. Substitution method

28. Substitution method

29. Substitution method

30. Substitution method
Remedy: Subtract off a lower-order term.

31. Recursion Tree
Use to generate a guess. Then verify by substitution method.
Consider the recurrence 𝑇 𝑛 =3𝑇 𝑛 4 +𝑐 𝑛 2

32. Recursion Tree Note: not a complete binary tree.
Depth for leftmost branch:
Subproblem size for a node at depth i is 𝑖 𝑛 Therefore, 𝑖 𝑛 = 1 → i = log3n
Depth for the rigtmost:
Subproblem size for a node at depth i is 𝑖 𝑛 .
Therefore, 𝑖 𝑛 = 1 → i = 𝑙𝑜𝑔 3/2 𝑛

33. Recursion Tree

34. Recursion Tree

35. Recursion Tree

36. Master Method
Let a, b, and k be integers satisfying a ≥ 1, b ≥ 2, and k ≥ 0
n/b can be either floor or ceiling function.
Floor: T(0) = u is given
Ceiling: T(1) = u is given
Examples: T(n) = 4T(n/2) + n  T(n)  ?
T(n) = 4T(n/2) + n2  T(n)  ?
T(n) = 4T(n/2) + n3  T(n)  ?

37. Master Method F(n) is polynomially smaller than 𝑛 log 𝑏 𝑎
F(n) is the same size as 𝑛 log 𝑏 𝑎
F(n) is polynomially larger than 𝑛 log 𝑏 𝑎 . It should also satisfy the regularity condition

38. Master Method F(n) is polynomially smaller than 𝑛 log 𝑏 𝑎
F(n) is the same size as 𝑛 log 𝑏 𝑎
F(n) is polynomially larger than 𝑛 log 𝑏 𝑎 . It should also satisfy the regularity condition

39. Master Method

40. Master Method