Divide and Conquer
•	We are going to use a restricted version of divide and conquer
•	The size problem will have to be divided into c parts each of size n/c
•	Tower of Hanoi is not divide and conquer, it will be called postponing the work because the size goes down to n-1, n-2,...1.
•	Divide and conquer examples: binary search, merge sort, quick sort.
•	Matrix multiplication
•	Traditional matrix multiplication O(n^3)
•	Has been brought down to O(n^2.376) by Winograd.
•	Known lower bound is O(n^2)
•	Study Strassen’s approach
•	See the board for regular calculations
•	15 = S1 = _a21 + _a22;    // addition 1
•	   12 = S2 =  S1  - _a11;    // addition 2
•	   -4 = S3 = _a11 - _a21;    // addition 3
•	   -8 = S4 = _a12 - S2;      // addition 4
•	   -1 = S5 = _b12 - _b11;    // addition 5
•	   4 = S6 = _b22 - S5;      // addition 6
•	   -3 = S7 = _b22 - _b12;    // addition 7
•	   -1 = S8 =  S6  - _b21;    // addition 8
•	
•	   48 = M1 =  S2  * S6;      // multiplication 1
•	   21 = M2 = _a11 * _b11;    // multiplication 2
•	   20 = M3 = _a12 * _b21;    // multiplication 3
•	   12 = M4 =  S3  * S7;      // multiplication 4
•	   - 15 = M5 =  S1  * S5;      // multiplication 5
•	   - 24 = M6 =  S4  * _b22;    // multiplication 6
•	   -8 = M7 = _a22 * S8;      // multiplication 7
•	69  = T1 = M1 + M2;                // addition 9
•	   81 = T2 = T1 + M4;                // addition 10
•	
•	   41 = C(1,1) = M2 + M3;            // addition 11
•	   30 = C(1,2) = T1 + M5 + M6;       // additions 12 and 13
•	   89 = C(2,1) = T2 - M7;            // addition 14
•	   66 = C(2,2) = T2 + M5;            // addition 15
•	The results are verified to be same (see the board)
•	We will only deal with square matrices and the number of rows will be power of two, i.e. 2, 4, 8, 16, 32
•	Any matrix with n > 2 will be broken into smaller matrices
•	e.g. 4*4 matrix will be converted to a 2*2 matrix, each elements is a 2*2 matrix.
•	See the board for example
•	T(n) = number of multiplications
•	T(n) = 7*T(n/2)+lower order
•	The time required for multiplying two matrices of size n involves 7 multiplication, each multiplication is multiplying two matrices of size n/2
•	T(n) = O(n^log 7)
•	Tower of Hanoi
•	T(n) = 2*T(n-1) + 1
•	Merge sort
•	T(n)=2*T(n/2)+n