1. Theory of Algorithms: Divide and Conquer

2. Objectives To introduce the divide-and-conquer mind set
To show a variety of divide-and-conquer solutions:
Merge Sort
Quick Sort
Convex Hull by Divide-and-Conquer
Strassen’s Matrix Multiplication
To discuss the strengths and weaknesses of a divide-and-conquer strategy

3. Divide-and-Conquer Best known algorithm design strategy:
Divide instance of problem into two or more smaller instances
Solve smaller instances recursively
Obtain solution to original (larger) instance by combining these solutions
Recurrence Templates apply
Recurrences are of the form
T(n) = aT(n/b) + f (n), a  1, b  2
Silly Example (Addition):
a0 + …. + an-1 = (a0 + … + an/2-1) + (an/2 + … an-1)

4. Divide-and-Conquer Illustrated
A PROBLEM OF
SIZE n
SUBPROBLEM 1
OF SIZE n/2
SUBPROBLEM 2
OF SIZE n/2
A SOLUTION TO
SUBPROBLEM 1
A SOLUTION TO
SUBPROBLEM 2
A SOLUTION TO THE
ORIGINAL PROBLEM

5. Mergesort Algorithm: Merging:
Split A[1..n] in half and put copy of each half into arrays B[1.. n/2 ] and C[1.. n/2 ]
Recursively MergeSort arrays B and C
Merge sorted arrays B and C into array A
Merging:
REPEAT until no elements remain in one of B or C
Compare 1st elements in the rest of B and C
Copy smaller into A, incrementing index of corresponding array
Once all elements in one of B or C are processed, copy the remaining unprocessed elements from the other array into A

6. Mergesort Example
6 4
7 2 1 6 4 7 2
4 6
2 7

7. Efficiency of Mergesort
Recurrence:
C(n) = 2 C(n/2) + Cmerge(n) for n > 1, C(1) = 0
Cmerge(n) = n - 1 in the worst case
All cases have same efficiency: (n log n)
Number of comparisons is close to theoretical minimum for comparison-based sorting:
log n! ≈ n lg n n
Space requirement: ( n ) (NOT in-place)
Can be implemented without recursion (bottom-up)
even if not analyzing in detail, show the recurrence for mergesort in worst case:
T(n) = 2 T(n/2) + (n-1) worst case comparisons for merge

8. Quicksort Select a pivot (partitioning element)
Rearrange the list into two sublists:
All elements positioned before the pivot are ≤ the pivot
Those positioned after the pivot are > the pivot
Requires a pivoting algorithm
Exchange the pivot with the last element in the first sublist
The pivot is now in its final position
QuickSort the two sublists p
A[i]≤p
A[i]>p

9. The Partition Algorithm

10. Quicksort Example 5 3 1 9 8 2 7 2 3 1 8 9 7 i j i j i j 5 3 1 9 8 2 7
i j
i j
i j
i j
i j
i j
i j
j i
j i
j i
Recursive Call Quicksort (2 3 1) & Quicksort (8 9 7)
Recursive Call Quicksort (1) & Quicksort (3)
Recursive Call Quicksort (7) & Quicksort (9)

11. Worst Case Efficiency of Quicksort
In the worst case all splits are completely skewed
For instance, an already sorted list!
One subarray is empty, other reduced by only one:
Make n+1 comparisons
Exchange pivot with itself
Quicksort left = Ø, right = A[1..n-1]
Cworst = (n+1) + n + … + 3 = (n+1)(n+2)/2 - 3 = (n2) p
A[i]≤p
A[i]>p

12. General Efficiency of Quicksort
Efficiency Cases:
Best: split in the middle — ( n log n)
Worst: sorted array! — ( n2)
Average: random arrays — ( n log n)
Improvements (in combination 20-25% faster):
Better pivot selection: median of three partitioning avoids worst case in sorted files
Switch to Insertion Sort on small subfiles
Elimination of recursion
Considered the method of choice for internal sorting for large files (n ≥ 10000)

13. Objectives To introduce the divide-and-conquer mind set
To show a variety of divide-and-conquer solutions:
Merge Sort
Quick Sort
Convex Hull by Divide-and-Conquer
Strassen’s Matrix Multiplication
To discuss the strengths and weaknesses of a divide-and-conquer strategy

14. QuickHull Algorithm Sort points by increasing x-coordinate values
Identify leftmost and rightmost extreme points P1 and P2 (part of hull)
Compute upper hull:
Find point Pmax that is farthest away from line P1P2
Quickhull the points to the left of line P1Pmax
Quickhull the points to the left of line PmaxP2
Similarly compute lower hull
Pmax L P2 L P1

15. Finding the Furthest Point
Given three points in the plane p1 , p2 , p3
Area of Triangle =  p1 p2 p3 = 1/2  D 
D =
 D  = x1 y2 + x3 y1 + x2 y3 - x3 y2 - x2 y1 - x1 y3
Properties of  D  :
Positive iff p3 is to the left of p1p2
Correlates with distance of p3 from p1p2 x1 y1 1 x2 y2 x3 y3

16. Efficiency of Quickhull
Finding point farthest away from line P1P2 is linear in the number of points
This gives same efficiency as quicksort:
Worst case: (n2)
Average case: (n log n)
If an initial sort is required, this can be accomplished in ( n log n) — no increase in asymptotic efficiency class
Alternative Divide-and-Conquer Convex Hull:
Graham’s scan and DCHull
Also ( n log n) but with lower coefficients

17. Strassen’s Matrix Multiplication
Strassen observed [1969] that the product of two matrices can be computed as follows:
Op Count:
Each of M1, …, M7 requires 1 mult and 1 or 2 add/sub
Total = 7 mul and 18 add/sub
Compared with brute force which requires 8 mults and 4 add/sub. But is asymptotic behaviour is NB
n/2  n/2 submatrices are computed recursively by the same method
C00
C01
C10
C11
A00
A01
A10
A11
B00
B01
B10
B11 = *
M1 + M4 -M5+M7
M3+ M5
M2 + M4
M1 + M3 - M2 + M6 =

18. Efficiency of Strassen’s Algorithm
If n is not a power of 2, matrices can be padded with zeros
Number of multiplications:
M(n) = 7 M(n/2) for n > 1, M(1) = 1
Set n = 2k, apply backward substitution
M(2k) = 7M(2k-1) = 7 [7 M(2k-2)] = 72 M(2k-2) = … = 7k
M(n) = 7log n = nlog 7 ≈ n2.807
Number of additions: A(n)  (nlog 7)
Other algorithms closer to the lower limit of n2 multiplications, but are even more complicated

19. Strengths and Weaknesses of Divide-and-Conquer
Generally improves on Brute Force by one base efficiency class
Easy to analyse using the Recurrence Templates
Weaknesses:
Often requires recursion, which introduces overheads
Can be inapplicable and inferior to simpler algorithmic solutions