ds 3Sorting.ppt

Bubble sort
Insertion sort
Selection sort

Sorting Algorithm
 Sorting takes an unordered collection and
makes it an ordered one.
2
5
12
35
42
77 101
5 12 35 42 77 101
1 2 3 4 5 6
1 2 3 4 5 6

"Bubbling Up" the Largest Element
 Traverse a collection of elements
 Move from the front to the end
 “Bubble” the largest value to the end using pair-
wise comparisons and swapping
3
5
12
35
42
77 101
1 2 3 4 5 6

4
5
12
35
42
77 101
Swap
42 77
1 2 3 4 5 6

5
5
12
35
77
42 101
Swap
35 77
1 2 3 4 5 6

6
5
12
77
35
42 101
Swap
12 77
1 2 3 4 5 6

7
5
77
12
35
42 101
No need to swap
1 2 3 4 5 6

8
5
77
12
35
42 101 Swap
5 101
1 2 3 4 5 6

9
77
12
35
42 5 101
Largest value correctly placed
1 2 3 4 5 6

Repeat “Bubble Up” How Many
Times?
 If we have N elements…
 And if each time we bubble an element, we
place it in its correct location…
 Then we repeat the “bubble up” process N –
1 times.
 This guarantees we’ll correctly
place all N elements.
10

“Bubbling” All the Elements
11
77
12
35
42 5 101
5
42
12
35 77 101
42
5
35
12 77 101
42
35
5
12 77 101
42
35
12
5 77 101
N
-
1
1 2 3 4 5 6

Bubble Sort
Algorithm
for i  1 to n-1 do
for j  1 to n-i do
if (A[j+1] < A[j]) swap A[j] and A[j+1] ;
}
}
Analysis:
In general, if the list has n elements, we will have to do
(n-1) + (n-2) …. + 2 +1 = (n-1) n / 2 comparisons.
=O(n^2)
12

Insertion Sort
INSERTION_SORT (A, N)
1. Set A[0] = -∞.
2. Repeat Step 3 to 5 for K = 2, 3, …, N:
3. Set TEMP = A[K] and PTR = K – 1.
4. Repeat while TEMP < A[PTR]:
(a) Set A[PTR+1] = A[PTR]
(b) Set PTR = PTR – 1.
[End of Loop.]
5. Set A[PTR+1] = TEMP.
[End of Loop 2.]
6. Return.

Insertion Sort Example
 Sort: 34 8 64 51 32 21
 34 8 64 51 32 21
 The algorithm sees that 8 is smaller than 34 so it swaps.
 8 34 64 51 32 21
 51 is smaller than 64, so they swap.
 8 34 51 64 32 21
 8 34 51 64 32 21 (from previous slide)
 The algorithm sees 32 as another smaller number and moves
it to its appropriate location between 8 and 34.
 8 32 34 51 64 21
 The algorithm sees 21 as another smaller number and moves
into between 8 and 32.
 Final sorted numbers:
 8 21 32 34 51 64

Insertion Sort Complexity
 This Sorting algorithm is frequently used when n is very small.
 Worst case occurs when array is in reverse order. The inner loop
must use K – 1 comparisons.
f(n) = 1 + 2 + 3 + ….+ (n – 1)
= n(n – 1)/2
= O(n2)
 In average case, there will be approximately (K – 1)/2 comparisons in
the inner loop.
f(n) = (1 + 2 + 3 + ….+ (n – 1))/2
= n(n – 1)/4
= O(n2)

Selection Sort
This algorithm sorts an array A with N elements.
SELECTION(A, N)
1. Repeat steps 2 and 3 for k=1 to N-1:
2. Call MIN(A, K, N, LOC).
3. [Interchange A[k] and A[LOC]]
Set Temp:= A[k], A[k]:= A[LOC] and A[LOC]:=Temp.
[End of step 1 Loop.]
4. Exit.
MIN(A, K, N, LOC).
1. Set MIN := A[K] and LOC:= K.
2. Repeat for j=k+1 to N:
If Min>A[j], then: Set Min:= A[j] and LOC:=J.
[End of if structure]
3. Return.

Selection Sort Example
1
3
2
9
6
4
8
8
3
2
9
6
4
1
8
3
4
9
6
2
1
8
6
4
9
3
2
1
8
9
6
4
3
2
1
8
6
9
4
3
2
1
9
8
6
4
3
2
1
9
8
6
4
3
2
1

Selection Sort Complexity
The number f(n) of comparisons in selection sort
algorithm is independent of original order of elements.
There are n-1 comparisons during pass 1 to find the
smallest element, n-2 comparisons during pass 2 to find
the second smallest element, and so on.
Accordingly,
f (n) = (n-1)+(n-2)+-----+2+1
= n(n-1)/2
= O(n2)
The f (n) holds the same value O(n2) both for worst case
and average case.

Comparing the Algorithms
Best Average Worst
Case Case Case
 Bubble Sort O(n) O(n2) O(n2)
 Insertion Sort O(n) O(n2) O(n2)
 Selection Sort O(n2) O(n2) O(n2)

ds 3Sorting.ppt

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