4 -1
Chapter 5
The Divide-and-Conquer Strategy
‫خوارزميات التقسيم والضم‬
4 -2
A simple example

finding the maximum of a set S of n numbers
4 -3
A general divide-and-conquer
algorithm
Step 1: If the problem size is small, solve this
problem directly; otherwise, split the original
problem into 2 sub-problems with equal sizes.
Step 2: Recursively solve these 2 sub-problems
by applying this algorithm.
Step 3: Merge the solutions of the 2 subproblems into a solution of the original
problem.
4 -4
Algorithm Merge-Sort
‫خوارزمية الترتيب بالدمج‬
Input: A set S of n elements.
Output: The sorted sequence of the inputs in
nondecreasing order.
Step 1: If |n|2, solve it directly.
Step 2: Recursively apply this algorithm to solve the
left half part and right half part of S, and the
results are stored in n1 and n2, respectively.
Step 3: Perform the two-way merge scheme on n1
and n2.
Mergesort
5
Algorithm:
 Split array A[1..n] in two and make copies of each
half in arrays B[1.. n/2 ] and C[1.. n/2 ]
arrays B and C‫ذ‬
Merge sorted arrays B and C into array A 
Pseudocode of Mergesort
Pseudocode of Merge
Time complexity: Θ(p+q) = Θ(n) comparisons
Merge
Sort
–
Example
Original Sequence
Sorted Sequence
18
26
32
6
43
15
9
1
1
6
9
15
18
26
32
6
43
15
9
1
6
18
26
18
26
32
6
43
15
9
1
18
26
6
32
18
26
32
6
43
15
9
1
26
32
6
18
26
32
6
43
15
9
1
18
Comp 122
32
18
26
1
9
15
43
32
43
15
43
43
43
1
9
15
9
1
Analysis of Merge Sort





Running time T(n) of Merge Sort:
Divide: computing the middle takes (1)
Conquer: solving 2 subproblems takes 2T(n/2)
Combine: merging n elements takes (n)
Total:
T(n) = (1)
if n = 1
T(n) = 2T(n/2) + (n)
if n > 1
 T(n) = (n lg n)
Comp 122
Quicksort
11




Select a pivot (partitioning element)
Rearrange the list so that all the elements in the positions
before the pivot are smaller than or equal to the pivot and
those after the pivot are larger than or equal to the pivot
Exchange the pivot with the last element in the first (i.e., ≤)
sublist – the pivot is now in its final position
Sort the two sublists recursively
p
A[i]≤p

A[i]p
Apply quicksort to sort the list 7 2 9 10 5 4
Design and Analysis of Algorithms – Chapter 5
pseudocode of quicksort:
ALGORITHM Quicksort(A[l..r])
//Sorts a subarray by quicksort
//Input: Subarray of array A[0..n − 1], defined by its left
and right
// indices l and r
//Output: Subarray A[l..r] sorted in nondecreasing order
if l < r
s ←Partition(A[l..r]) //s is a split position
Quicksort(A[l..s − 1])
Quicksort(A[s + 1..r])
Quicksort Example
13

Recursive implementation with the left most array
entry selected as the pivot element.
Design and Analysis of Algorithms – Chapter 5
Exercise :


Apply quicksort to sort the list E, X, A,M, P, L, E in
alphabetical order.
Draw the tree of the recursive calls made.