CS20L-Lecture Set 11 - Binary Heap

8/7/2019 CS20L-Lecture Set 11 - Binary Heap

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Binary HeapA 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 areadded from top to bottom, left to right, withoutleaving any spaces. A binary tree is completely full ifit is of height, h, and has 2h+1-1 nodes.

Heapness

The item in the tree with the highest priority is at thetop of the tree, and the same is true for everysubtree.


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Binary HeapBinary tree of height, h, is complete iff

it is empty

or

its left subtree is complete of height h-1and its right subtree is completely full ofheight h-2

or

its left subtree is completely full of heighth-1 and its right subtree is complete of

height h-1.


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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 thechild(ren).


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Binary HeapTo build a heap, data is placed in the tree as it arrives.

Algorithm:

1. add a new node at the next leaf position2. use this new node as the current position

3. 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


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Binary HeapNew nodes are always added on the deepest levelin 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 aspartially ordered.

A given set of data can be formed into many different

heaps (depends on the order in which the data arrives.)


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Binary Heap

Example:

Data arrives to be heaped in the order:

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

31


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Binary Heap

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

54 54

87

87

54

87

54 27

87

54 27

67

87

67 27

54

87

67 27

54 19


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Binary Heap

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

27

87

67

54 19

27

87

67

54 19 31


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Binary Heap

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

29

31

87

67

54 19 27 29

31

87

67

54 19 27 29

18

etc


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Binary HeapTo 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

CS20L-Lecture Set 11 - Binary Heap

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