1. Divide and Conquer Neil Tang 4/24/2008
CS223 Advanced Data Structures and Algorithms

2. CS223 Advanced Data Structures and Algorithms
Course Survey
Please complete the course survey by May 3 (Sat) at:
CS223 Advanced Data Structures and Algorithms

3. CS223 Advanced Data Structures and Algorithms
Class Overview
Basic idea
Quick sort
Merge sort
Integer multiplication
Matrix multiplication
CS223 Advanced Data Structures and Algorithms

4. CS223 Advanced Data Structures and Algorithms
Basic Idea
Divide: divide the original problem to several sub-problems.
Conquer: Solve the sub-problems recursively.
Combine: Combine the solutions to sub-problems to form a solution for the original problem.
CS223 Advanced Data Structures and Algorithms

5. CS223 Advanced Data Structures and Algorithms
Merge Sort
Divide: Divide the N-element sequence into 2 subsequences of N/2 each.
Conquer: Sort each subsequence recursively using merge sort.
Combine: Merge two sorted subsequences to produce a single sorted sequence.
The time complexity is O(NlogN)
CS223 Advanced Data Structures and Algorithms

6. CS223 Advanced Data Structures and Algorithms
Merge Sort
CS223 Advanced Data Structures and Algorithms

7. CS223 Advanced Data Structures and Algorithms
Merge Sort
T(N) = 2T(N/2) + N
a=2, b=2, f(N)=N
log22 = 1
So we have case 2, T(N) = (NlogN).
CS223 Advanced Data Structures and Algorithms

8. CS223 Advanced Data Structures and Algorithms
Quick Sort
Divide: Divide the sequence into 2 subsequences, s.t. each element in the 1st subsequence is less than or equal to each element in the 2nd subsequence.
Conquer: Sort each subsequence recursively using quick sort.
Combine: no work is needed.
CS223 Advanced Data Structures and Algorithms

9. CS223 Advanced Data Structures and Algorithms
Quick Sort
age 25
people
age  25
age  25 23
age < 23 30
age  30
age  23
age < 30
CS223 Advanced Data Structures and Algorithms

10. CS223 Advanced Data Structures and Algorithms
Quick Sort
T(N) = T(N-1) + N
The Master method does not work.
T(N) = O(N2)
CS223 Advanced Data Structures and Algorithms

11. Integer Multiplication
Compute XY, where X and Y are N-digit integers.
Divide: X=XL104+XR , Y=YL104+YR
Conquer and combine:
XY = XLYL108 + (XLYR+ XRYL)104+XRYR
CS223 Advanced Data Structures and Algorithms

12. Integer Multiplication
T(N) = 4T(N/2) + (N)
a=4, b=2, f(N)= (N)
log24 = 2 
So we have case 1, T(N) = (N2).
CS223 Advanced Data Structures and Algorithms

13. CS223 Advanced Data Structures and Algorithms
A Better Algorithm
CS223 Advanced Data Structures and Algorithms

14. Integer Multiplication
T(N) = 3T(N/2) + (N)
a=3, b=2, f(N)= (N)
log23 = 1.59 
So we have case 1, T(N) = (N1.59).
CS223 Advanced Data Structures and Algorithms

15. Matrix Multiplication
Compute AB, where A and B are NN matrix.
Divide: Divide each matrix to 4 quadrants.
Conquer and combine:
CS223 Advanced Data Structures and Algorithms

16. Matrix Multiplication
CS223 Advanced Data Structures and Algorithms

17. Matrix Multiplication
T(N) = 8T(N/2) + (N2)
a=8, b=2, f(N)= (N2)
log28 = 3 
So we have case 1, T(N) = (N3).
CS223 Advanced Data Structures and Algorithms

18. CS223 Advanced Data Structures and Algorithms
A Better Algorithm
CS223 Advanced Data Structures and Algorithms

19. Matrix Multiplication
T(N) = 7T(N/2) + (N2)
a=7, b=2, f(N)= (N2)
log27 = 2.81 
So we have case 1, T(N) = (N2.81).
CS223 Advanced Data Structures and Algorithms