Binary Heap

1
Binary Heap

• A special kind of binary tree. It has two
• properties that are not generally true for other
• trees
• Completeness
• The tree is complete, which means that nodes are
  added from top to bottom, left to right, without
  leaving any spaces. A binary tree is completely
  full if it is of height, h, and has 2h1-1 nodes.
  
• Heapness
• The item in the tree with the highest priority
  is at the top of the tree, and the same is true
  for every subtree.

2
Binary Heap

• Binary tree of height, h, is complete iff
• it is empty
• or
• its left subtree is complete of height h-1 and
  its right subtree is completely full of height
  h-2
• or
• its left subtree is completely full of height
  h-1 and its right subtree is complete of
• height h-1.

3
Binary Heap

• In simple terms
• A heap is a binary tree in which every parent is
  greater than its child(ren).
• A Reverse Heap is one in which the rule is every
  parent is less than the child(ren).

4
Binary Heap

• To build a heap, data is placed in the tree as it
  arrives.
• Algorithm
• add a new node at the next leaf position
• use this new node as the current position
• While new data is greater than that in the parent
  of the current node
• move the parent down to the current node
  
• make the parent (now vacant) the current
  node
• Place data in current node

5
Binary Heap

• New nodes are always added on the deepest level
• in an orderly way.
• The tree never becomes unbalanced.
• There is no particular relationship among the
  data items
• in the nodes on any given level, even the ones
  that have
• the same parent
• A heap is not a sorted structure. Can be regarded
  as
• partially ordered.
• A given set of data can be formed into many
  different
• heaps (depends on the order in which the data
  arrives.)

6
Binary Heap

• Example
• Data arrives to be heaped in the order
• 54, 87, 27, 67, 19, 31, 29, 18, 32, 56, 7, 12,
• 31

7
Binary Heap

• 54, 87, 27, 67, 19, 31, 29, 18, 32, 56, 7, 12, 31

54
8
Binary Heap

• 54, 87, 27, 67, 19, 31, 29, 18, 32, 56, 7, 12, 31

31
9
Binary Heap

• 54, 87, 27, 67, 19, 31, 29, 18, 32, 56, 7, 12, 31

29
18
etc
10
Binary Heap

• To delete an element from the heap
• Algorithm
• If node is a leaf, delete it
• Else if node has a right child, locate the
  deepest leaf of right subtree and replace node
  with that leaf. Then re-heapify
• Else replace node with the left child.

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Binary Heap
Example Deleting the root node, T
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Binary Heap
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Binary Heap
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Binary Heap