Heaps, MinHeap, MaxHeap,
BinaryHeap, Whatever…
CMPS 2133
Terry Griffin
Binary Heap

Definition:
•
•

Property
•

A Tree is heap ordered if the key in each node is larger
than or equal to the keys in all of it’s children
Equivalently, the key in each node if a heap-ordered tree
is smaller than or equal to the key of its parent
No node in a heap ordered tree has a key larger than the
key at the root
Definition:
•
A heap is a set of nodes with keys arranged in a
complete heap-ordered binary tree, represented as an
array.
Binary Heap: Properties
Properties.
•
•
Min (or max) element is in root.
Heap with N elements has height = log2 N.
06
N = 14
Height = 3
14
78
83
45
18
91
81
47
77
84
53
99
64
Binary Heaps: Array Implementation
Implementing
•
binary heaps.
Use an array: no need for explicit parent or child
pointers.
Parent(i) = i/2
 Left(i)
= 2i
 Right(i)
= 2i + 1

06
1
14
45
2
3
78
18
47
53
4
5
6
7
83
91
81
77
84
99
64
8
9
10
11
12
13
14
Binary Heap: Insertion
Insert
•
•
element x into heap.
Insert into next available slot.
Bubble up until it's heap ordered.

Peter principle: nodes rise to level of incompetence
06
14
78
83
45
18
91
81
47
77
84
53
99
64
42
next free slot
Binary Heap: Insertion
Insert
•
•
element x into heap.
Insert into next available slot.
Bubble up until it's heap ordered.

Peter principle: nodes rise to level of incompetence
swap with parent
06
14
78
83
45
18
91
81
47
77
84
53
99
64
42
Binary Heap: Insertion
Insert
•
•
element x into heap.
Insert into next available slot.
Bubble up until it's heap ordered.

Peter principle: nodes rise to level of incompetence
swap with parent
06
14
78
83
45
18
91
81
47
77
84
42
99
64
53
42
Binary Heap: Insertion
Insert
•
•
•
element x into heap.
Insert into next available slot.
Bubble up until it's heap ordered.
O(log N) operations.
stop: heap ordered
06
14
78
83
42
18
91
81
47
77
84
45
99
64
53
Binary Heap: Decrease Key
Decrease
•
•
key of element x to k.
Bubble up until it's heap ordered.
O(log N) operations.
06
14
78
83
42
18
91
81
47
77
84
45
99
64
53
Binary Heap: Delete Min
Delete
•
•
minimum element from heap.
Exchange root with rightmost leaf.
Bubble root down until it's heap ordered.
06
14
78
83
42
18
91
81
47
77
84
45
99
64
53
Binary Heap: Delete Min
Delete
•
•
minimum element from heap.
Exchange root with rightmost leaf.
Bubble root down until it's heap ordered.
53
14
78
83
42
18
91
81
47
77
84
45
99
64
06
Binary Heap: Delete Min
Delete
•
•
minimum element from heap.
Exchange root with rightmost leaf.
Bubble root down until it's heap ordered.
exchange with left child
53
14
78
83
42
18
91
81
47
77
84
45
99
64
Binary Heap: Delete Min
Delete
•
•
minimum element from heap.
Exchange root with rightmost leaf.
Bubble root down until it's heap ordered.
exchange with right child
14
53
78
83
42
18
91
81
47
77
84
45
99
64
Binary Heap: Delete Min
Delete
•
•
•
minimum element from heap.
Exchange root with rightmost leaf.
Bubble root down until it's heap ordered.
O(log N) operations.
stop: heap ordered
14
18
78
83
42
53
91
81
47
77
84
45
99
64
Binary Heap Operations

Heapify
Binary Heap Operations
Heapify Example
Binary Heap Operations
Heapify Example
Binary Heap Operations
Heapify Example
Binary Heap Operations
Heapify Example
Binary Heap Operations
Heapify Example
Binary Heap Operations
Heapify Example
Binary Heap Operations
Heapify Example
Binary Heap Operations
Heapify


Aside from the recursive call, what is the
running time of Heapify.
How many time can Heapify recursively call
itself.
Binary Heap Operations
Building a Heap
You can build a heap from the bottom up by
running Heapify on successive sub-arrays.
 For an array of length n, all elements in the
range: A[ceiling(n/2) + 1 … n ] are heaps. Why?
 Order of processing guarantees that the
children of node i are heaps when that node is
processed.

Binary Heap Operations
Building a Heap
Binary Heap Operations
Heap-Insert
Heap-Insert(A,key)
1.
heap-size[A]<-heap-size[A]+1
2.
i<-heap-size[A]
3.
while i>1 and A[Parent(i)]<key
4.
5.
6.
do A[i]<-A[Parent(i)]
i<-Parent(i)
A[i]<-key
Binary Heap Operations
Heap-Insert
Binary Heap Operations
Heap-Insert
Binary Heap Operations
Heap-Insert
Binary Heap Operations
Heap-Insert
Binary Heap Operations
Heap Extract Max
Heap-Extract Max
1.
if heap-size[A] < 1
2.
then error
3.
max<- A[1]
4.
A[1]<-A[heap-size[A]]
5.
heap-size[A]<-heap-size[A] - 1
6.
Heapify(A,1)
7.
return max
Binary Heap Operations
Heap Extract Max example
Binary Heap Operations
Heap Extract Max example
Binary Heap Operations
Heap Extract Max example
Binary Heap Operations
Heap Extract Max example
Binary Heap Operations
Heap Extract Max example