lec 6 feb 17, 2011 section 2.5 of text (review of heap) chapter 3
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Lec 6 Feb 17, 2011
Section 2.5 of text (review of heap)
Chapter 3
Review of heap (Sec 2.5)
Heap is a data structure that supports a priority queue. Two versions (Max-heap, min-heap)
Max-heap operations (can do in O(log n) time. Insert(H, x) – add x to H. Delete-max(H) – remove the max elt. from H. Other operations: increase-key, decrease-key, delete(j)
– delete the key stored in index j of heap etc. Operations that take O(n) time:
search(x), delete(x) etc.
Min Heap with 9 Nodes
Complete binary tree with 9 nodes.
Min Heap With 9 Nodes
Min-heap property: A[k] <= A[2*k] (if 2*k <= n) and A[k] <= A[2*k+1] (if 2*k+1 <= n).
2
4
6 7 9 3
8 6
3
Max Heap With 9 Nodes
Example of a Max-heap
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
Moving Up And Down A Heap
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8
6 7 2 6
5 1
7
1
2 3
4 5 6 7
8 9Parent of node with index k is k/2
Left child of a node with index j is 2*jRight child of a node with index j is 2*j + 1
Putting An Element Into A Max Heap
Place to add the new key
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8
6 7 2 6
5 1
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7
Putting An Element Into A Max Heap
Example: New element is 5.
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6 7 2 6
5 1
7
75
Putting An Element Into A Max Heap
New element is 20.
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2 6
5 1
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7
7
Putting An Element Into A Max Heap
New element is 20.
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2 6
5 1
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77
Putting An Element Into A Max Heap
New element is 20.
9
86
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2 6
5 1
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77
Putting An Element Into A Max Heap
New element is 20.
9
86
7
2 6
5 1
7
77
20
Putting An Element Into A Max Heap
Complete binary tree with 11 nodes.
9
86
7
2 6
5 1
7
77
20
Putting An Element Into A Max Heap
New element is 15.
9
86
7
2 6
5 1
7
77
20
Putting An Element Into A Max Heap
New element is 15.
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8
6
7
2 6
5 1
7
77
20
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Putting An Element Into A Max Heap
New element is 15.
8
6
7
2 6
5 1
7
77
20
8
9
15
Complexity of insert
Complexity is O(log n), where n is heap size.
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2 6
5 1
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77
20
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9
15
DeleteMax operation
Max element is in the root.
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2 6
5 1
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77
20
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9
15
DeleteMax
After max element is removed.
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2 6
5 1
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77 8
9
15
DeleteMax
Heap with 10 nodes. Location needs to be vacated. Find the right place to reinsert 8.
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2 6
5 1
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77 8
9
15
DeleteMax
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2 6
5 1
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77
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15
Reinsert 8 into the heap.
DeleteMax
Reinsert 8 into the heap.
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2 6
5 1
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77
9
15
DeleteMax
Reinsert 8 into the heap.
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2 6
5 1
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77
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15
8
DeleteMax – Another example
Max element is 15.
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2 6
5 1
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77
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15
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DeleteMax – Ex 2
After max element is removed.
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2 6
5 1
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77
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DeleteMax – Ex 2
Heap with 9 nodes.
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2 6
5 1
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77
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8
DeleteMax – Ex 2
Reinsert 7.
6 2 6
5 1
79
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DeleteMax – Ex 2
Reinsert 7.
6 2 6
5 1
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DeleteMax – Ex 2
Reinsert 7.
6 2 6
5 1
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9
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Complexity of DeleteMax
• Complexity is O(log n).
• Involves working down the heap, two comparisons and 1 assignment per level. There are at most log2 (n+1) levels.
• Total complexity <= 3 log2 (n+1) = O(log n).
6 2 6
5 1
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Delete a key at a given index
Want an algorithm of complexity O(log n).
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5 1
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Delete (2)
Delete a key at a given index
To perform Delete(j):
A[j] = A[size]; size--;
adjust the heap at position j;
How to adjust?
6 2 6
5 1
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Delete (2)
similar to DeleteMax
Delete a key at a given index : Ex – 2
16 2 6
15
7
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Delete (6)
Adjustment may require percolate_up or percolate_down
Augmenting a heap
Suggest a data structure that acts like both a min-heap and max-heap.
i.e., it should support all three operations in O(log n) time:• Insert• DeleteMin• DeleteMax
Any suggestion?