Five basic algorithm design methods:

1
Chapter 10Algorithm Design Techniques

• Five basic algorithm design methods
• Greedy Method
• Divide and Conquer
• Dynamic Programming
• Backtracking
• Branch and Bound

2
Chapter 10Divide and Conquer

• Three Step Approach
• 1. Divide problem into two or more smaller
  instances
• 2. Solve each of these smaller instances
• 3. Combine the solutions to obtain the solution
  for the larger problem
• Example Detecting a counterfeit coin
• Bag with 16 coins
• One of them may be counterfeit
• Counterfeit coins are lighter
• Determine whether bag contains a counterfeit coin

3
Chapter 10Divide and Conquer

• Three Step Approach
• 1. Divide problem into two or more smaller
  instances
• 2. Solve each of these smaller instances
• 3. Combine the solutions to obtain the solution
  for the larger problem
• Example Gold Nuggets
• Given 8 nuggets, find the heaviest
• Find the lightest

4
Chapter 10Divide and Conquer

• Matrix Multiplication
• 1 2 5 5 6 1 15 22 59 16 21
  52
• 4 1 3 X 2 1 5 45 12 39 46
  11 32
• 3 5 2 9 2 4 35 52 29 36 51
  22
• C(i, j)

5
Chapter 10Divide and Conquer

• n/2 n/2
• n/2
• n/2

6
Chapter 10Divide and Conquer

• C1 A1 B1 A2 B2
• C2 A1 B2 A2 B4
• C3 A3 B1 A4 B3
• C4 A3 B2 A4 B4
• Eight multiplications, four additions

7
Chapter 10Divide and Conquer

• Review of Insertion Sort

templateltclass Tgt void InsertionSort(T a, int
n) for (int i 1 i lt n i) //insert
ai into a0i-1 T t ai int j for
(j i - 1 j gt 0 t lt aj j--) aj1
aj aj 1 t
8
Chapter 10Divide and Conquer

• Merge Sort

• Partition elements into 2 or more subcollections
• Sort each
• Combine the solution
• Example 4 2 7 10 9 1 8 5 3
• How to partition? How to sort?
• One possibility
• First n-1 elements in one subcollection,
• Last element in the second
• A B
• 2 7 10 9 1 8 5 3
• But this is just a recursive version of
  InsertionSort().O(n) calls to Sort(), O(n) merges

9
Chapter 10Divide and Conquer

• Merge Sort

• Divide and conquer approach divide the list
  into two equal groups. This avoids the O(n2)
  complexity
• 4 6 3 8 2 5 7
• 10 4 6 3 8 2 5 7
• 10 4 6 3 8 2 5 7
• 10 4 6 3 8
  2 5 7
• Then merge the groups.

10
Chapter 10Divide and Conquer

• Merge Sort

templateltclass Tgt void MergeSort(T a, int
n) T b new T n int s 1 while
(sltn) MergePass(a, b, s, n) ss Merge
Pass(b,a,s,n) ss
11
Chapter 10Divide and Conquer

• Merge Sort

templateltclass Tgt void MergePass(T x, T y,
int s, int n) int i 0 while (i lt n - 2s)
Merge(x, y, i, i s - 1, i 2s - 1) i
i 2s if (i s lt n) Merge(x, y, i,
is-1, n-1) else for (int j i j lt n 1
j) yj xj
12
Chapter 10Divide and Conquer

• Merge Sort

templateltclass Tgt void Merge(T c, T d, int l,
int m, int r) int i l, j m1, k
l while ((i lt m) (j lt r)) if (ci lt
cj) dk ci else dk
cj if (i gt m) for (int q j q ltr
q) dk cq else for (int q i q lt
m q) dk cq
13
Chapter 10Divide and Conquer

• Quick Sort

• n elements to be sorted
• Partition into 3 groups
• left middle right
• Exactly one element in the middle The pivot,
  or index, or partitioning element.
• Select an element from a0n-1 for middle. This
  is the pivot
• Partition the remaining elements into segments
  left and right. Partition s.t. no element in the
  left is gt pivot and no element in the right is lt
  pivot.
• Sort left using QuickSort recursively
• Sort right using QuickSort recursively
• The answer is left, middle, right

14
Chapter 10Divide and Conquer

templateltclass Tgt void QuickSort(T a, int n)
quickSort(a, 0, n-1) templateltclass Tgt void
quickSort(T a, int l, int r) if (l gt r)
return int i l, j r1 T pivot
ai while (true) do i i 1 while
(ai lt pivot) do j j - 1 while (aj gt
pivot) if (i gt j) break Swap(ai,
aj) al aj aj pivot quickSort
(a, l, j-1) quickSort(a, j1, r)
15
Chapter 10Divide and Conquer

• Quick Sort

• n elements to be sorted
• Partition into 3 groups
• left middle right
• Exactly one element in the middle The pivot,
  or index, or partitioning element.
• Select an element from a0n-1 for middle. This
  is the pivot
• Partition the remaining elements into segments
  left and right. Partition s.t. no element in the
  left is gt pivot and no element in the right is lt
  pivot.
• Sort left using QuickSort recursively
• Sort right using QuickSort recursively
• The answer is left, middle, right