Construct min heap from the given sequence

Sep 1, 2020Download as PPTX, PDF0 likes50 views

The document discusses two methods for constructing a min-heap from a given sequence of numbers. The first method builds a complete binary tree and then checks that each node's value is greater than its parent's value, swapping if needed. The second method inserts nodes one by one, checking and swapping with the parent if the parent's value is greater. Both methods are walked through step-by-step using the example sequence 15, 3, 6, 18, 5, 9, 11.

DATA STRUCTURES
AND ALGORITHMS
MIN-HEAP
CONSTRUCTION
SADIA ZAR
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 1

MIN-HEAP – DATA STRUCTURE
• Construct a min-heap from the following sequence:
15, 3, 6, 18 ,5 ,9 ,11
• In this lecture I’ve discussed two methods to construct a min-heap.
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
2

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
In the first method, at first we will construct a complete
binary tree first and then we will check the condition i.e
A[Parent[i]] < = A[i] for every node in the tree.
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
3

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
4

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
5

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
6

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
7

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
8

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
9

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
10

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
Now Let’s check the condition for every node
Starting from leaf nodes
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
11

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
Now Let’s check the condition for every node
Starting from leaf nodes
i = 3
Parent[i] = Parent[3] = 1
So the condition is A[1] <= A[3]
i.e, 3 <= 18 so no need to swap
0
1
2
3
4 5 6
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
12

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
Now Let’s check the condition for every node
Starting from leaf nodes
i = 4
Parent[i] = Parent[4] = 1
So the condition is A[1] <= A[4]
i.e, 3 <= 5 so no need to swap
0
1
2
3
4 5 6
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
13

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
Now Let’s check the condition for every node
Starting from leaf nodes
i = 5
Parent[i] = Parent[5] = 2
So the condition is A[2] <= A[5]
i.e, 6 <= 9 so no need to swap
0
1
2
3
4 5 6
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
14

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
Now Let’s check the condition for every node
Starting from leaf nodes
i = 6
Parent[i] = Parent[6] = 2
So the condition is A[2] <= A[6]
i.e, 6 <= 11 so no need to swap
0
1
2
3
4 5 6
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
15

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
Now Let’s check the condition for every node
Starting from leaf nodes
i = 1
Parent[i] = Parent[1] = 0
So the condition is A[0] <= A[1]
i.e, 15 <= 3 (false) so we have to swap
0
1
2
3
4 5 6
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
16

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
Now Let’s check the condition for every node
Starting from leaf nodes
After swapping check other nodes as well
i = 2
Parent[i] = Parent[2] = 0
So the condition is A[0] <= A[2]
i.e, 3 <= 6 so no need to swap
0
1
2
3
4 5 6
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
17

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
Now Let’s check the condition for every node
Starting from leaf nodes
Again check all the nodes now the problem
Lies in node 1 and node 4
i = 4
Parent[i] = Parent[4] = 1
So the condition is A[1] <= A[4]
i.e, 15 <= 5 (false) so we have to swap
0
1
2
3
4 5 6
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
18

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
Now Let’s check the condition for every node
Starting from leaf nodes
Now again check
i = 4
Parent[i] = Parent[4] = 1
So the condition is A[1] <= A[4]
i.e, 5 <= 15 so no need to swap
0
1
2
3
4 5 6
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
19

MIN-HEAP – DATA STRUCTURE
FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11
Now Let’s check the condition for every node
Starting from leaf nodes
As you can see that the min-heap is ready
0
1
2
3
4 5 6
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
20

MIN-HEAP – DATA STRUCTURE
SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11
In the second method, as we construct a complete binary
tree, we will check the condition i.e A[Parent[i]] < = A[i] for
every node that is inserted in the tree.
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
21

MIN-HEAP – DATA STRUCTURE
SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11
0
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
22

MIN-HEAP – DATA STRUCTURE
SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11
i = 1
Parent[i] = Parent[1] = 0
Check the condition A[0]<=A[1]
15<=3 (false) so, we need to swap these two values
0
1
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
23

MIN-HEAP – DATA STRUCTURE
SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11
i = 1
Parent[i] = Parent[1] = 0
Check the condition A[0]<=A[1]
3<=15 (true) so, continue
0
1
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
24

MIN-HEAP – DATA STRUCTURE
SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11
i = 2
Parent[i] = Parent[2] = 0
Check the condition A[0]<=A[2]
i.e, 3<=15 so no need to swap
0
1 2
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
25

MIN-HEAP – DATA STRUCTURE
SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11
i = 3
Parent[i] = Parent[3] = 1
Check the condition A[1]<=A[3]
i.e, 15<=18 so no need to swap
0
1 2
3
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
26

MIN-HEAP – DATA STRUCTURE
SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11
i = 4
Parent[i] = Parent[4] = 1
Check the condition A[1]<=A[4]
i.e, 15<=5 (false) so we have to
swap
0
1 2
3 4
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
27

MIN-HEAP – DATA STRUCTURE
SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11
No the min-heap is okay
0
1 2
3 4
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
28

MIN-HEAP – DATA STRUCTURE
SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11
i = 5
Parent[i] = Parent[5] = 2
Check the condition A[2]<=A[5]
i.e, 6<=9 so no need to swap
0
1 2
3 4 5
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
29

MIN-HEAP – DATA STRUCTURE
SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11
i = 6
Parent[i] = Parent[6] = 2
Check the condition A[2]<=A[6]
i.e, 6<=11 so no need to swap
0
1 2
3 4 5 6
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
30

MIN-HEAP – DATA STRUCTURE
SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11
Now Let’s check the condition for every node
Starting from leaf nodes
As you can see that the min-heap is ready
0
1
2
3
4 5 6
GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W),
RAWALPINDI.
31

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Construct min heap from the given sequence

1. DATA STRUCTURES AND ALGORITHMS MIN-HEAP CONSTRUCTION SADIA ZAR GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 1

2. MIN-HEAP – DATA STRUCTURE • Construct a min-heap from the following sequence: 15, 3, 6, 18 ,5 ,9 ,11 • In this lecture I’ve discussed two methods to construct a min-heap. GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 2

3. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 In the first method, at first we will construct a complete binary tree first and then we will check the condition i.e A[Parent[i]] < = A[i] for every node in the tree. GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 3

4. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 4

5. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 5

6. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 6

7. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 7

8. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 8

9. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 9

10. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 10

11. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 Now Let’s check the condition for every node Starting from leaf nodes GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 11

12. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 Now Let’s check the condition for every node Starting from leaf nodes i = 3 Parent[i] = Parent[3] = 1 So the condition is A[1] <= A[3] i.e, 3 <= 18 so no need to swap 0 1 2 3 4 5 6 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 12

13. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 Now Let’s check the condition for every node Starting from leaf nodes i = 4 Parent[i] = Parent[4] = 1 So the condition is A[1] <= A[4] i.e, 3 <= 5 so no need to swap 0 1 2 3 4 5 6 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 13

14. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 Now Let’s check the condition for every node Starting from leaf nodes i = 5 Parent[i] = Parent[5] = 2 So the condition is A[2] <= A[5] i.e, 6 <= 9 so no need to swap 0 1 2 3 4 5 6 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 14

15. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 Now Let’s check the condition for every node Starting from leaf nodes i = 6 Parent[i] = Parent[6] = 2 So the condition is A[2] <= A[6] i.e, 6 <= 11 so no need to swap 0 1 2 3 4 5 6 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 15

16. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 Now Let’s check the condition for every node Starting from leaf nodes i = 1 Parent[i] = Parent[1] = 0 So the condition is A[0] <= A[1] i.e, 15 <= 3 (false) so we have to swap 0 1 2 3 4 5 6 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 16

17. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 Now Let’s check the condition for every node Starting from leaf nodes After swapping check other nodes as well i = 2 Parent[i] = Parent[2] = 0 So the condition is A[0] <= A[2] i.e, 3 <= 6 so no need to swap 0 1 2 3 4 5 6 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 17

18. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 Now Let’s check the condition for every node Starting from leaf nodes Again check all the nodes now the problem Lies in node 1 and node 4 i = 4 Parent[i] = Parent[4] = 1 So the condition is A[1] <= A[4] i.e, 15 <= 5 (false) so we have to swap 0 1 2 3 4 5 6 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 18

19. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 Now Let’s check the condition for every node Starting from leaf nodes Now again check i = 4 Parent[i] = Parent[4] = 1 So the condition is A[1] <= A[4] i.e, 5 <= 15 so no need to swap 0 1 2 3 4 5 6 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 19

20. MIN-HEAP – DATA STRUCTURE FIRST METHOD - 15, 3, 6, 18 ,5 ,9 ,11 Now Let’s check the condition for every node Starting from leaf nodes As you can see that the min-heap is ready 0 1 2 3 4 5 6 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 20

21. MIN-HEAP – DATA STRUCTURE SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11 In the second method, as we construct a complete binary tree, we will check the condition i.e A[Parent[i]] < = A[i] for every node that is inserted in the tree. GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 21

22. MIN-HEAP – DATA STRUCTURE SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11 0 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 22

23. MIN-HEAP – DATA STRUCTURE SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11 i = 1 Parent[i] = Parent[1] = 0 Check the condition A[0]<=A[1] 15<=3 (false) so, we need to swap these two values 0 1 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 23

24. MIN-HEAP – DATA STRUCTURE SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11 i = 1 Parent[i] = Parent[1] = 0 Check the condition A[0]<=A[1] 3<=15 (true) so, continue 0 1 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 24

25. MIN-HEAP – DATA STRUCTURE SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11 i = 2 Parent[i] = Parent[2] = 0 Check the condition A[0]<=A[2] i.e, 3<=15 so no need to swap 0 1 2 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 25

26. MIN-HEAP – DATA STRUCTURE SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11 i = 3 Parent[i] = Parent[3] = 1 Check the condition A[1]<=A[3] i.e, 15<=18 so no need to swap 0 1 2 3 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 26

27. MIN-HEAP – DATA STRUCTURE SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11 i = 4 Parent[i] = Parent[4] = 1 Check the condition A[1]<=A[4] i.e, 15<=5 (false) so we have to swap 0 1 2 3 4 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 27

28. MIN-HEAP – DATA STRUCTURE SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11 No the min-heap is okay 0 1 2 3 4 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 28

29. MIN-HEAP – DATA STRUCTURE SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11 i = 5 Parent[i] = Parent[5] = 2 Check the condition A[2]<=A[5] i.e, 6<=9 so no need to swap 0 1 2 3 4 5 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 29

30. MIN-HEAP – DATA STRUCTURE SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11 i = 6 Parent[i] = Parent[6] = 2 Check the condition A[2]<=A[6] i.e, 6<=11 so no need to swap 0 1 2 3 4 5 6 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 30

31. MIN-HEAP – DATA STRUCTURE SECOND METHOD - 15, 3, 6, 18 ,5 ,9 ,11 Now Let’s check the condition for every node Starting from leaf nodes As you can see that the min-heap is ready 0 1 2 3 4 5 6 GOVERNMENT VIQAR-UN-NISA POST GRADUATE COLLEGE (W), RAWALPINDI. 31

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Construct min heap from the given sequence

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Construct min heap from the given sequence