Sorting Algorithms: Insertion Sort and Complexity Analysis, Slides of Design and Analysis of Algorithms

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Insertion, Bubble AND Selection

Sort

Sorting Algorithms

2

Some Definitions

• Internal Sort
  • The data to be sorted is all stored in the computer’s main memory.
• External Sort
  • Some of the data to be sorted might be stored in some external, slower, device.
• In Place Sort
  • The amount of extra space required to sort the data is constant with the input size.

4

To insert 12, we need to make room for it by moving first 36 and then 24.

Insertion Sort

5

Insertion Sort

7

Insertion Sort

input array

left sub-array (^) right sub-array

at each iteration, the array is divided in two sub-arrays:

sorted unsorted

8

Insertion Sort

10

Loop Invariant for Insertion Sort

Alg.: INSERTION-SORT (A)

for j ← 2 to n do key ← A[ j ] Insert A[ j ] into the sorted sequence A[1.. j -1] i ← j - 1 while i > 0 and A[i] > key do A[i + 1] ← A[i] i ← i – 1 A[i + 1] ← key

Invariant : at the start of the for loop the elements in A[1.. j-1] are in sorted order

11

Analysis of Insertion Sort

cost times

c 1 n c 2 n- 0 n- c 4 n- c 5 c 6 c 7 c 8 n-

 (^) 

n j 2 tj  (^)  

n j 2 ( tj^1 )  (^)  

n j 2 ( tj^1 )

( ) ( 1 ) ( 1 ) ^1 ^ ^1 ^8 ( 1 ) 2

7 2

6 2

 1  2   4   5            

T n cn c n c n c t c t c t c n

n

j

j

n

j

j

n

j

j

INSERTION-SORT (A)

for j ← 2 to n do key ← A[ j ] Insert A[ j ] into the sorted sequence A[1.. j -1] i ← j - 1 while i > 0 and A[i] > key do A[i + 1] ← A[i] i ← i – 1 A[i + 1] ← key tj: # of times the while statement is executed at iteration j

13

Worst Case Analysis

• The array is in reverse sorted order
  • Always A[i] > key in while loop test
  • Have to compare key with all elements to the left of the j - th position  compare with j-1 elements  tj = j

a quadratic function of n

• T(n) = (n^2 ) order of growth in n^2

1 2 2

( 1) ( 1) (^1) ( 1) ( 1) 2 2 2

n n n j j j j n n^ j n n^ j n n     ^ ^ ^  ^ ^ ^ ^  ^  

( ) 1 2 ( 1 ) 4 ( 1 ) 5 ( 2 1 ) (^1)  6 ( 2 ^1 ) 7 ( 2 ^1 ) 8 (  1 )    

T n  cn  c n   c n   c ^ n n   c n n c n n c n

 an^2  bn  c

“while i > 0 and A[i] > key”

( ) ( 1 ) ( 1 )  1   1  8 ( 1 ) 2

7 2

6 2

 1  2   4   5            

T n cn c n c n c t c t c t c n

n j

j

n j

j

n j

j

using (^) we have:

14

Comparisons and Exchanges in Insertion Sort

INSERTION-SORT (A)

for j ← 2 to n

do key ← A[ j ] Insert A[ j ] into the sorted sequence A[1.. j -1]

i ← j - 1

while i > 0 and A[i] > key do A[i + 1] ← A[i] i ← i – 1 A[i + 1] ← key

cost times

c 1 n c 2 n- 0 n- c 4 n- c 5

c 6 c 7 c 8 n-

 (^) 

n j 2 tj  (^)  

n j 2 ( tj^1 )  (^)  

n j 2 ( tj^1 )

 n^2 /2 comparisons

 n^2 /2 exchanges

16

Bubble Sort

• Idea:
  • Repeatedly pass through the array
  • Swaps adjacent elements that are out of order
• Easier to implement, but slower than Insertion

sort

1 2 3 n

i

8 4 6 9 2 3 1 j

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Example

8 4 6 9 2 3 1 i = 1 j

8 4 6 9 2 1 3 i = 1 j

8 4 6 9 1 2 3 i = 1 j

8 4 6 1 9 2 3 i = 1 j

8 4 1 6 9 2 3 i = 1 j

8 1 4 6 9 2 3 i = 1 j

1 8 4 6 9 2 3 i = 1 j

1 8 4 6 9 2 3 i = 2 j 1 2 8 4 6 9 3 i = 3 j 1 2 3 8 4 6 9 i = 4 j 1 2 3 4 8 6 9 i = 5 j 1 2 3 4 6 8 9 i = 6 j 1 2 3 4 6 8 9 i = 7 j Docsity.com

19

Bubble-Sort Running Time

Thus,T(n) = (n^2 )

2 2 1 1 1

n n n

i i i

n n n n

where n i n i n

  

 ^ ^  ^  ^ ^ ^ 

Alg.: BUBBLESORT(A) for i  1 to length[A] do for j  length[A] downto i + 1 do if A[j] < A[j -1] then exchange A[j]  A[j-1]

T(n) = c 1 (n+1) +    



n

i

n i 1

c 2 (^1 ) c 3 



 

n

i

n i 1

(^ ) c 4 





n

i

n i 1

( )

= (n) + (c 2 + c 2 + c 4 ) 





n

i

n i 1

( )

Comparisons:  n^2 /

Exchanges:  n^2 /

c 1 c 2 c 3 c 4

20

Selection Sort

• Idea:
  • Find the smallest element in the array
  • Exchange it with the element in the first position
  • Find the second smallest element and exchange it with the element in the second position
  • Continue until the array is sorted
• Disadvantage:
  • Running time depends only slightly on the amount of order in the file