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Priority Queues. Two kinds of priority queues: Min priority queue. Max priority queue. Complexity Of Operations. Sorted List Insert O(N) Delete O(1) Unsorted List Insert O(1) Delete O(N). Heaps. Two key properties Heap shape Complete binary tree Value at node - PowerPoint PPT Presentation
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Priority Queues
Two kinds of priority queues:
• Min priority queue.
• Max priority queue.
Complexity Of Operations
• Sorted List– Insert O(N)– Delete O(1)
• Unsorted List– Insert O(1)– Delete O(N)
Heaps• Two key properties
– Heap shape• Complete binary tree
– Value at node• Min Heap: Smaller than or equal to values in subtrees
• Max Heap: Larger than or equal to values in subtrees
• Example heap– X Y– X Z
Y
X
Z
Heap & Non-heap Examples
Heaps Non-heaps
6
2
22
8 45 25
6
2
22
8 45 25
8
6 45
5
6 22
25
5
5 45
5
Min Heap With 9 Nodes
Complete binary tree with 9 nodes that is also a min tree.
2
4
6 7 9 3
8 6
3
Max Heap With 9 Nodes
Complete binary tree with 9 nodes that is also a max tree.
9
8
6 7 2 6
5 1
7
Heap Height
Since a heap is a complete binary tree, the height of an n node heap is log2 (n+1).
9 8 7 6 7 2 6 5 1
1 2 3 4 5 6 7 8 9 100
A Heap Is Efficiently Represented As An Array
9
8
6 7 2 6
5 1
7
Inserting An Element Into A Max Heap
Complete binary tree with 10 nodes.
9
8
6 7 2 6
5 1
7
7
Inserting An Element Into A Max Heap
New element is 5.
9
8
6 7 2 6
5 1
7
75
Inserting An Element Into A Max Heap
New element is 20.
9
8
6
7
2 6
5 1
7
7
7
Inserting An Element Into A Max Heap
New element is 20.
9
8
6
7
2 6
5 1
7
77
Inserting An Element Into A Max Heap
New element is 20.
9
86
7
2 6
5 1
7
77
Inserting An Element Into A Max Heap
New element is 20.
9
86
7
2 6
5 1
7
77
20
Inserting An Element Into A Max Heap
Complete binary tree with 11 nodes.
9
86
7
2 6
5 1
7
77
20
Inserting An Element Into A Max Heap
New element is 15.
9
86
7
2 6
5 1
7
77
20
Inserting An Element Into A Max Heap
New element is 15.
9
8
6
7
2 6
5 1
7
77
20
8
Inserting An Element Into A Max Heap
New element is 15.
8
6
7
2 6
5 1
7
77
20
8
9
15
Heap Operations – Insert( X )
• Algorithm– Add X to end of tree– While (X > parent)
• Swap X with parent // X bubbles up tree
• Complexity– # of swaps proportional to height of tree– O( log(n) )
Insertion
Insert(int key){ n++; //heap size+1 i=n; while (key > heap[i/2] && i>1) { heap[i]=heap[i/2]; i=i/2; } heap[i]=key;}
Removing The Max Element
Max element is in the root.
8
6
7
2 6
5 1
7
77
20
8
9
15
Removing The Max Element
After max element is removed.
8
6
7
2 6
5 1
7
77 8
9
15
Removing The Max Element
Heap with 10 nodes.
8
6
7
2 6
5 1
7
77 8
9
15
Reinsert 8 into the heap.
Removing The Max Element
Reinsert 8 into the heap.
6
7
2 6
5 1
7
77
9
15
Removing The Max Element
Reinsert 8 into the heap.
6
7
2 6
5 1
7
77
9
15
Removing The Max Element
Reinsert 8 into the heap.
6
7
2 6
5 1
7
77
9
15
8
Removing The Max Element
Max element is 15.
6
7
2 6
5 1
7
77
9
15
8
Removing The Max Element
After max element is removed.
6
7
2 6
5 1
7
77
9
8
Removing The Max Element
Heap with 9 nodes.
6
7
2 6
5 1
7
77
9
8
Removing The Max Element
Reinsert 7.
6 2 6
5 1
79
8
Removing The Max Element
Reinsert 7.
6 2 6
5 1
7
9
8
Removing The Max Element
Reinsert 7.
6 2 6
5 1
7
9
8
7
DeletionDelete(){ key=heap[1]; data=heap[n--]; i=1; while ((j=2*i)<=n) { if (2*i+1<=n) { if (heap[2*i+1] > heap[2*i]) j=2*i+1; } if (heap[j] > data); { heap[i]=heap[j]; i=j; } else break; } heap[i]=data; return key;}
Complexity Of Remove Max Element
Complexity is O(log n).
6 2 6
5 1
7
9
8
7
Binary Search Tree20
10
6
2 8
15
40
30
25
Only keys are shown.
Definition Of Binary Search Tree
• A binary tree.
• Each node has a (key, value) pair.
• For every node x, all keys in the left subtree of x are smaller than that in x.
• For every node x, all keys in the right subtree of x are greater than that in x.
Binary search trees
• Average depth of a node is O(log N); maximum depth of a node is O(N)
Two binary search trees representing
the same set:
The Operation Insert()20
10
6
2 8
15
40
30
25
Insert a pair whose key is 35.
35
The Operation Insert()
Insert a pair whose key is 7.
20
10
6
2 8
15
40
30
25 35
7
The Operation Delete()
Four cases:
No element with delete key.
Element is in a leaf.
Element is in a degree 1 node.
Element is in a degree 2 node.
Delete From A Leaf
Delete a leaf element. key = 7
20
10
6
2 8
15
40
30
25 35
7
18
Delete From A Degree 1 Node
Delete from a degree 1 node. key = 40
20
10
6
2 8
15
40
30
25 35
7
18
Delete From A Degree 1 Node (contd.)
Delete from a degree 1 node. key = 15
20
10
6
2 8
15
40
30
25 35
7
18
Delete From A Degree 2 Node
Delete from a degree 2 node. key = 10
20
10
6
2 8
15
40
30
25 35
7
18
Delete From A Degree 2 Node20
10
6
2 8
15
40
30
25
Replace with largest key in left subtree (or smallest in right subtree).
35
7
18
Delete From A Degree 2 Node20
10
6
2 8
15
40
30
25
Replace with largest key in left subtree (or smallest in right subtree).
35
7
18
Delete From A Degree 2 Node20
8
6
2 8
15
40
30
25
Replace with largest key in left subtree (or smallest in right subtree).
35
7
18
Delete From A Degree 2 Node20
8
6
2 8
15
40
30
25
Largest key must be in a leaf or degree 1 node.
35
7
18
Another Delete From A Degree 2 Node
Delete from a degree 2 node. key = 20
20
10
6
2 8
15
40
30
25 35
7
18
Delete From A Degree 2 Node20
10
6
2 8
15
40
30
25
Replace with largest in left subtree.
35
7
18
Delete From A Degree 2 Node20
10
6
2 8
15
40
30
25
Replace with largest in left subtree.
35
7
18
Delete From A Degree 2 Node18
10
6
2 8
15
40
30
25
Replace with largest in left subtree.
35
7
18
Delete From A Degree 2 Node18
10
6
2 8
15
40
30
25
Complexity is O(height).
35
7
Indexed Binary Search Tree
• Binary search tree.
• Each node has an additional field. leftSize = number of nodes in its left subtree +
1
Example Indexed Binary Search Tree20
10
6
2 8
15
40
30
25 35
7
18
1
1 2
2
5
1
1
8
1 1
2
4
leftSize values are in red
Search by rankint SBR(node* root, int rank){ if (rank>treesize) return; ptr=root; while (1) { if (ptr->index==rank) return ptr->data; else if (ptr->index < rank) { ptr=ptr->rightchild; rank=rank - ptr->index; } else ptr=ptr->leftchild; }}
