A Heap Is Efficiently Represented As An Array

1
A Heap Is Efficiently Represented As An Array
9
8
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1
1
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10
0
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Moving Up And Down A Heap
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Inserting An Element Into A Max Heap

• Complete binary tree with 10 nodes.

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Inserting An Element Into A Max Heap
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1
7
5

• New element is 5.

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Inserting An Element Into A Max Heap
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1
7

• New element is 20.

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Inserting An Element Into A Max Heap
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7

• New element is 20.

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Inserting An Element Into A Max Heap
9
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1
7
7

• New element is 20.

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Inserting An Element Into A Max Heap
20
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7
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1
7
7

• New element is 20.

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Inserting An Element Into A Max Heap
20
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6

7
5
1
7
7

• Complete binary tree with 11 nodes.

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Inserting An Element Into A Max Heap
20
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6

7
5
1
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7

• New element is 15.

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Inserting An Element Into A Max Heap
20
9
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1
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7

• New element is 15.

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Inserting An Element Into A Max Heap
20
7
15
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2
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1
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7

• New element is 15.

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Complexity of Inserting

• Complexity is O(log n), where n is heap size.

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Deleting the Max Element

• Max element is in the root.

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Deleting the Max Element
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7

• After max element is removed.

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Deleting the Max Element
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1
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7

• Heap with 10 nodes.

Reinsert 8 into the heap.
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Deleting the Max Element
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15
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1
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7

• Reinsert 8 into the heap.

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Deleting the Max Element
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7

• Reinsert 8 into the heap.

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Deleting the Max Element
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1
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7

• Reinsert 8 into the heap.

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Deleting the Max Element
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1
7
7

• Max element is 15.

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Deleting the Max Element
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1
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7

• After max element is removed.

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Deleting the Max Element
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1
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7

• Heap with 9 nodes.

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Deleting the Max Element
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1

• Reinsert 7.

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Deleting the Max Element
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1

• Reinsert 7.

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Deleting the Max Element
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1

• Reinsert 7.

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Complexity of Remove Max Element

• Complexity is O(log n).

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Initializing a Max Heap

• input array -, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
  11

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Initializing a Max Heap
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• Start at rightmost array position that has a
  child.

Index is n/2.
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Initializing a Max Heap
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• Move to next lower array position.

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Initializing a Max Heap
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Initializing a Max Heap
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Initializing a Max Heap
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Initializing a Max Heap
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Initializing a Max Heap
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Initializing a Max Heap
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Find a home for 2.
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Initializing a Max Heap
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Find a home for 2.
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Initializing a Max Heap
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2
Done, move to next lower array position.
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Initializing a Max Heap
1
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Find home for 1.
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Initializing a Max Heap
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Find home for 1.
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Initializing a Max Heap
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Find home for 1.
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Initializing a Max Heap
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Find home for 1.
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Initializing a Max Heap
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1
Done.
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Time Complexity
11
7
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2
1
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Height of heap h. Number of subtrees with root
at level j is lt 2 j-1. Time for each subtree is
O(h-j1).
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Complexity
Time for level j subtrees is lt 2j-1(h-j1)
t(j). Total time is t(1) t(2) t(h-1)
O(n).
45
Leftist Trees

• Linked binary tree.
• Can do everything a heap can do and in the same
  asymptotic complexity.
• Can meld two leftist tree priority queues in
  O(log n) time.

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Extended Binary Trees

• Start with any binary tree and add an external
  node wherever there is an empty subtree.
• Result is an extended binary tree.

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A Binary Tree
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An Extended Binary Tree
number of external nodes is n1
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The Function s()

• For any node x in an extended binary tree, let
  s(x) be the length of a shortest path from x to
  an external node in the subtree rooted at x.

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s() Values Example
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s() Values Example
2
2
1
2
1
1
0
1
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1
1
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0
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Properties of s()

• If x is an external node, then s(x) 0.
• Otherwise,
• s(x) min s(leftChild(x)),
• s(rightChild(x)) 1

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Height Biased Leftist Trees

• A binary tree is a (height biased) leftist tree
  iff for every internal node x, s(leftChild(x)) gt
  s(rightChild(x))

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A Leftist Tree
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Leftist Trees--Property 1

• In a leftist tree, the rightmost path is a
  shortest root to external node path and the
  length of this path is s(root).

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A Leftist Tree
2
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Length of rightmost path is 2.
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Leftist TreesProperty 2

• The number of internal nodes is at least
• 2s(root) - 1
• Because levels 1 through s(root) have no external
  nodes.
• So, s(root) lt log(n1)

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A Leftist Tree
2
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1
1
0
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0
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Levels 1 and 2 have no external nodes.
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Leftist TreesProperty 3

• Length of rightmost path is O(log n), where n is
  the number of nodes in a leftist tree.
• Follows from Properties 1 and 2.

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Leftist Trees as Priority Queues
Min leftist tree leftist tree that is a min
tree. Used as a min priority queue. Max leftist
tree leftist tree that is a max tree. Used as a
max priority queue.
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A Min Leftist Tree
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Some Min Leftist Tree Operations
insert() delete() meld() initialize() insert()
and delete() use meld().
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Insert Operation

• insert(7)

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Insert Operation

• insert(7)

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Create a single node min leftist tree.
7
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Insert Operation

• insert(7)

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Create a single node min leftist tree. Meld the
two min leftist trees.
7
66
Delete Min
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Delete Min
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Delete the root.
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Delete Min
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Delete the root. Meld the two subtrees.
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Meld Two Min Leftist Trees
Traverse only the rightmost paths so as to get
logarithmic performance.
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Meld Two Min Leftist Trees
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6
Meld right subtree of tree with smaller root and
all of other tree.
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Meld Two Min Leftist Trees
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Meld right subtree of tree with smaller root and
all of other tree.
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Meld Two Min Leftist Trees
6
4
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8
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6
Meld right subtree of tree with smaller root and
all of other tree.
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Meld Two Min Leftist Trees
6
8
Meld right subtree of tree with smaller root and
all of other tree. Right subtree of 6 is empty.
So, result of melding right subtree of tree with
smaller root and other tree is the other tree.
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Meld Two Min Leftist Trees
Make melded subtree right subtree of smaller root.
Swap left and right subtree if s(left) lt s(right).
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Meld Two Min Leftist Trees
Make melded subtree right subtree of smaller root.
Swap left and right subtree if s(left) lt s(right).
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Meld Two Min Leftist Trees
Make melded subtree right subtree of smaller root.
Swap left and right subtree if s(left) lt s(right).
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Meld Two Min Leftist Trees
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Initializing in O(n) Time

• Create n single node min leftist trees and place
  them in a FIFO queue
• Repeatedly remove two min leftist trees from the
  FIFO queue, meld them, and put the resulting min
  leftist tree into the FIFO queue
• The process terminates when only 1 min leftist
  tree remains in the queue
• Analysis is the same as for heap initialization