A heap is a binary tree A heap is best implemented in sequential representation using an array Two important uses of heaps are i efficient implementation of priority queues ii sorting Heapsort ID: 577297
Download Presentation The PPT/PDF document "1 Heaps" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
1
Heaps
A heap is a binary tree.
A heap is best implemented in sequential representation (using an array).
Two important uses of heaps are:
(i) efficient implementation of priority queues
(ii) sorting -- Heapsort.Slide2
2
A Heap
10
20
30
40
50
25
55
52
42
Any node’s key value is
less than its children’s.Slide3
3
Sequential Representation of Trees
There are three methods of representing a binary tree using array representation.
1. Using index values to represent edges:
class Node {
Data
elt
; int
left; int
right; } Node;
Node[] BinaryTree[TreeSize
]; Slide4
4
Method 1: Example
A
D
B
C
E
G
I
Index
Element
Left
Right
1
A
2
3
2
B
4
6
3
C
0
0
4
D
0
0
5
I
0
0
6
E
7
0
7
G
5
0Slide5
5
Method 2
2. Store the nodes in one of the natural traversals:
class Node {
Data
elt
;
boolean left;
boolean right; };
Node[] BinaryTree[TreeSize
]; Slide6
6
Method 2: Example
A
D
B
C
E
G
I
Index
Element
Left
Right
1
A
T
T
2
B
T
T
3
D
F
F
4
E
T
F
5
G
T
F
6
I
F
F
7
C
F
F
Elements stored in Pre-Order traversalSlide7
7
Method 3
3. Store the nodes in fixed positions: (i) root goes into first index, (ii) in general left child of tree[i] is stored in tree[2i] and right child in tree[2i+1].Slide8
8
Method 3: Example
A
D
B
C
E
G
A
B
C
D
E
-
-
-
-
G
-
1
2
3
4
5
6
7
8
9
10
11
12Slide9
9
Heaps
Heaps are represented sequentially using the third method.
Heap is a
complete binary tree
: shortest-path length tree with nodes on the lowest level in their leftmost positions.
Complete Binary Tree:
let h
be the height of the heapfor i =
0, … , h - 1, there are 2i
nodes of depth iat depth h
- 1, the internal nodes are to the left of the external nodesSlide10
Heaps (Cont.)
Max-Heap has
max
element as root. Min-Heap has
min
element as root.
The elements in a heap satisfy heap conditions: for Min-Heap: key[parent] < key[left-child] or key[right-child].The
last node of a heap is the rightmost node of maximum depth
10
2
6
5
7
9
last nodeSlide11
11
Heap: An example
10
25
2
0
3
0
4
0
42
50
52
55
[1]
10
10
10
[2]
20
30
40
[3]
25
20
20
[4]
30
40
50
[5]
40
50
42
[6]
42
25
30
[7]
50
55
25
[8]
52
52
52
[9]
55
42
55
All the three arrangements
satisfy min heap conditionsSlide12
12
Heap: An example
10
20
30
40
50
25
55
52
42
[1]
10
10
10
[2]
20
30
40
[3]
25
20
20
[4]
30
40
50
[5]
40
50
42
[6]
42
25
30
[7]
50
55
25
[8]
52
52
52
[9]
55
42
55
All the three arrangements
satisfy min heap conditionsSlide13
13
Heap: An example
10
20
40
50
42
30
25
52
55
[1]
10
10
10
[2]
20
30
40
[3]
25
20
20
[4]
30
40
50
[5]
40
50
42
[6]
42
25
30
[7]
50
55
25
[8]
52
52
52
[9]
55
42
55
All the three arrangements
satisfy min heap conditionsSlide14
14
Constructing Heaps
There are two methods of constructing heaps:
Using SiftDown operation.
Using SiftUp operation.
SiftDown operation inserts a new element into the Heap from the top.
SiftUp operation inserts a new element into the Heap from the bottom.Slide15
15
ADT Heap
Elements:
The elements are called
HeapElements
.
Structure: The elements of the heap satisfy the heap conditions.
Domain: Bounded. Type name: Heap.Slide16
16
ADT Heap
Operations:
Method
SiftUp (int n)
requires: Elements H[1],H[2],…,H[n-1] satisfy heap conditions. results:
Elements H[1],H[2],…,H[n] satisfy heap conditions.Method SiftDown (int m,n)
requires: Elements H[m+1],H[m+2],…,H[n] satisfy the heap conditions. results: Elements H[m],H[m+1],…,H[n] satisfy the heap conditions.
Method Heap (int n) // Constructor results
: Elements H[1],H[2],….H[n] satisfy the heap conditions.Slide17
ADT Heap: Element
public class
HeapElement
<T> {
T data;
Priority p;
public
HeapElement(T e, Priority pty) {
data = e; p =
pty; }
// Setters/Getters...}
17Slide18
Heaps
18
Insertion into a Heap
Method
insertItem
of the priority queue ADT corresponds to the insertion of a key
k
to the heap
The insertion algorithm consists of three stepsFind the insertion node z (the new last node)
Store k at z
Restore the heap-order property (discussed next)© 2010 Goodrich, TamassiaSlide19
Heaps
19
Upheap
After the insertion of a new key
k
, the heap-order property may be violated
Algorithm
upheap restores the heap-order property by swapping
k along an upward path from the insertion nodeUpheap terminates when the key
k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log
n), upheap runs in O
(log n) time
© 2010 Goodrich, Tamassia
2
6
5
7
9
insertion node
zSlide20
Heaps
20
Upheap
After the insertion of a new key
k
, the heap-order property may be violated
Algorithm
upheap restores the heap-order property by swapping
k along an upward path from the insertion nodeUpheap terminates when the key
k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log
n), upheap runs in O
(log n) time
© 2010 Goodrich, Tamassia
2
6
5
7
9
1Slide21
Heaps
21
Upheap
After the insertion of a new key
k
, the heap-order property may be violated
Algorithm
upheap restores the heap-order property by swapping
k along an upward path from the insertion nodeUpheap terminates when the key
k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log
n), upheap runs in O
(log n) time
© 2010 Goodrich, Tamassia
2
1
5
7
9
6Slide22
Heaps
22
Upheap
After the insertion of a new key
k
, the heap-order property may be violated
Algorithm
upheap restores the heap-order property by swapping
k along an upward path from the insertion nodeUpheap terminates when the key
k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log
n), upheap runs in O
(log n) time
© 2010 Goodrich, Tamassia
1
2
5
7
9
6Slide23
Heaps
23
Upheap
After the insertion of a new key
k
, the heap-order property may be violated
Algorithm
upheap restores the heap-order property by swapping
k along an upward path from the insertion nodeUpheap terminates when the key
k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log
n), upheap runs in O
(log n) time
© 2010 Goodrich, Tamassia
1
2
5
7
9
6
insertion node
zSlide24
Heaps
24
Upheap
After the insertion of a new key
k
, the heap-order property may be violated
Algorithm
upheap restores the heap-order property by swapping
k along an upward path from the insertion nodeUpheap terminates when the key
k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log
n), upheap runs in O
(log n) time
© 2010 Goodrich, Tamassia
1
2
5
7
9
6
10Slide25
Heaps
25
Upheap
After the insertion of a new key
k
, the heap-order property may be violated
Algorithm
upheap restores the heap-order property by swapping
k along an upward path from the insertion nodeUpheap terminates when the key
k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log
n), upheap runs in O
(log n) time
© 2010 Goodrich, Tamassia
1
2
5
7
9
6
10
insertion node
zSlide26
Heaps
26
Upheap
After the insertion of a new key
k
, the heap-order property may be violated
Algorithm
upheap restores the heap-order property by swapping
k along an upward path from the insertion nodeUpheap terminates when the key
k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log
n), upheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
1
2
5
7
9
6
10
4Slide27
Heaps
27
Upheap
After the insertion of a new key
k
, the heap-order property may be violated
Algorithm
upheap restores the heap-order property by swapping
k along an upward path from the insertion nodeUpheap terminates when the key
k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log
n), upheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
1
2
5
7
4
6
10
9Slide28
Heaps
28
Upheap
After the insertion of a new key
k
, the heap-order property may be violated
Algorithm
upheap restores the heap-order property by swapping
k along an upward path from the insertion nodeUpheap terminates when the key
k reaches the root or a node whose parent has a key smaller than or equal to k
Since a heap has height O(log
n), upheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
1
2
4
7
5
6
10
9Slide29
Heaps
29
Removal from a Heap (§ 7.3.3)
Method removeMin of the priority queue ADT corresponds to the removal of the root key from the heap
The removal algorithm consists of three steps
Replace the root key with the key of the last node
w
Remove
w Restore the heap-order property (discussed next)
2
6
5
7
9
last node
w
7
6
5
9
w
new last node
© 2010 Goodrich, TamassiaSlide30
Heaps
30
Downheap
After replacing the root key with the key
k
of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key
k
along a downward path from the rootUpheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to
k Since a heap has height O
(log n), downheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
2
6
5
7
9
last node
wSlide31
Heaps
31
Downheap
After replacing the root key with the key
k
of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key
k
along a downward path from the rootUpheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to
k Since a heap has height O
(log n), downheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
7
6
5
7
9
last
node
wSlide32
Heaps
32
Downheap
After replacing the root key with the key
k
of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key
k
along a downward path from the rootUpheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to
k Since a heap has height O
(log n), downheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
7
6
5
9
delete last
node
wSlide33
Heaps
33
Downheap
After replacing the root key with the key
k
of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key
k
along a downward path from the rootUpheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to
k Since a heap has height O
(log n), downheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
7
6
5
9
DownHeap
/
SiftDownSlide34
Heaps
34
Downheap
After replacing the root key with the key
k
of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key
k
along a downward path from the rootUpheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to
k Since a heap has height O
(log n), downheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
5
6
7
9Slide35
Heaps
35
Downheap
After replacing the root key with the key
k
of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key
k
along a downward path from the rootUpheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to
k Since a heap has height O
(log n), downheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
5
6
7
9
last
node
wSlide36
Heaps
36
Downheap
After replacing the root key with the key
k
of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key
k
along a downward path from the rootUpheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to
k Since a heap has height O
(log n), downheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
9
6
7
9
last
node
wSlide37
Heaps
37
Downheap
After replacing the root key with the key
k
of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key
k
along a downward path from the rootUpheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to
k Since a heap has height O
(log n), downheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
9
6
7
delete last
node
wSlide38
Heaps
38
Downheap
After replacing the root key with the key
k
of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key
k
along a downward path from the rootUpheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to
k Since a heap has height O
(log n), downheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
9
6
7
DownHeap
/
SiftDownSlide39
Heaps
39
Downheap
After replacing the root key with the key
k
of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key
k
along a downward path from the rootUpheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to
k Since a heap has height O
(log n), downheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
6
9
7Slide40
Heaps
40
Downheap
After replacing the root key with the key
k
of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key
k
along a downward path from the rootUpheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to
k Since a heap has height O
(log n), downheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
6
9
7
last
node
wSlide41
Heaps
41
Downheap
After replacing the root key with the key
k
of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key
k
along a downward path from the rootUpheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to
k Since a heap has height O
(log n), downheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
9
9
7
last
node
wSlide42
Heaps
42
Downheap
After replacing the root key with the key
k
of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key
k
along a downward path from the rootUpheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to
k Since a heap has height O
(log n), downheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
9
7
delete last
node
wSlide43
Heaps
43
Downheap
After replacing the root key with the key
k
of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key
k
along a downward path from the rootUpheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to
k Since a heap has height O
(log n), downheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
9
7
DownHeap
/
SiftDownSlide44
Heaps
44
Downheap
After replacing the root key with the key
k
of the last node, the heap-order property may be violated
Algorithm downheap restores the heap-order property by swapping key
k
along a downward path from the rootUpheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to
k Since a heap has height O
(log n), downheap runs in
O(log n) time
© 2010 Goodrich, Tamassia
7
9Slide45
Heaps
45
Updating the Last Node
The insertion node can be found by traversing a path of
O
(log
n
)
nodesGo up until a left child or the root is reachedIf a left child is reached, go to the right child
Go down left until a leaf is reachedSimilar algorithm for updating the last node after a removal
© 2010 Goodrich, TamassiaSlide46
Heaps
46
Heap-Sort
Consider a priority queue with
n
items implemented by means of a heap
the space used is
O
(n)methods
insert and removeMin take O
(log n) time
methods size, isEmpty, and min
take time O(1) time
Using a heap-based priority queue, we can sort a sequence of
n elements in O(
n log n
) time
The resulting algorithm is called heap-sortHeap-sort is much faster than quadratic sorting algorithms, such as insertion-sort and selection-sort
© 2010 Goodrich, TamassiaSlide47
Heaps
47
Vector-based Heap Implementation
We can represent a heap with
n
keys by means of a vector of length
n
+
1
For the node at rank ithe left child is at rank 2
ithe right child is at rank 2
i +
1Links between nodes are not explicitly storedThe cell of at rank 0 is not used
Operation insert corresponds to inserting at rank n +
1Operation removeMin corresponds to removing at rank n
Yields in-place heap-sort
2
6
5
7
9
2
5
6
9
7
1
2
3
4
5
0
© 2010 Goodrich, TamassiaSlide48
Heaps
48
We can construct a heap storing
n
given keys in using a bottom-up construction with
log
n
phasesIn phase i
, pairs of heaps with 2i -
1 keys are merged into heaps with 2i
+1-1 keys
Bottom-up Heap Construction
2
i
-
1
2
i -
1
2
i
+
1
-
1
© 2010 Goodrich, TamassiaSlide49
Heaps
49
Example
15
16
12
4
7
6
20
23
25
15
16
5
12
4
11
7
6
27
20
23
© 2010 Goodrich, TamassiaSlide50
Heaps
50
Example (contd.)
25
15
16
5
12
4
11
9
6
27
20
23
15
25
16
4
12
5
6
9
11
23
20
27
© 2010 Goodrich, TamassiaSlide51
Heaps
51
Example (contd.)
7
15
25
16
4
12
5
8
6
9
11
23
20
27
4
15
25
16
5
12
7
6
8
9
11
23
20
27
© 2010 Goodrich, TamassiaSlide52
Heaps
52
Example (end)
4
15
25
16
5
12
7
10
6
8
9
11
23
20
27
5
15
25
16
7
12
10
4
6
8
9
11
23
20
27
© 2010 Goodrich, TamassiaSlide53
Heaps
53
Merging Two Heaps
We are given two two heaps and a key
k
We create a new heap with the root node storing
k
and with the two heaps as subtrees
We perform downheap to restore the heap-order property © 2010 Goodrich, Tamassia
3
5
8
2
6
4Slide54
Heaps
54
Merging Two Heaps
We are given two two heaps and a key
k
We create a new heap with the root node storing
k
and with the two heaps as subtrees
We perform downheap to restore the heap-order property
7
3
5
8
2
6
4
© 2010 Goodrich, Tamassia
kSlide55
Heaps
55
Merging Two Heaps
We are given two two heaps and a key
k
We create a new heap with the root node storing
k
and with the two heaps as subtrees
We perform downheap to restore the heap-order property
7
3
5
8
2
6
4
© 2010 Goodrich, Tamassia
k
MergeSlide56
Heaps
56
Merging Two Heaps
We are given two two heaps and a key
k
We create a new heap with the root node storing
k
and with the two heaps as subtrees
We perform downheap to restore the heap-order property
7
3
5
8
2
6
4
© 2010 Goodrich, Tamassia
Downheap
/
SiftDownSlide57
Heaps
57
Merging Two Heaps
We are given two two heaps and a key
k
We create a new heap with the root node storing
k
and with the two heaps as subtrees
We perform downheap to restore the heap-order property
2
3
5
8
7
6
4
© 2010 Goodrich, Tamassia
Downheap
/
SiftDownSlide58
Heaps
58
Merging Two Heaps
We are given two two heaps and a key
k
We create a new heap with the root node storing
k
and with the two heaps as subtrees
We perform downheap to restore the heap-order property
2
3
5
8
4
6
7
© 2010 Goodrich, Tamassia
Downheap
/
SiftDownSlide59
Heaps
59
Heaps and Priority Queues
We can use a heap to implement a priority queue
We store a (key, element) item at each internal node
We keep track of the position of the last node
(2, Sue)
(6, Mark)
(5, Pat)
(9, Jeff)
(7, Anna)
© 2010 Goodrich, TamassiaSlide60
60
Priority Queue as Heap
Representation as a Heap
public class
HeapPQ
<T> {
Heap<T>
pq
;
/** Creates a new instance of HeapPQ */
public HeapPQ
() { pq
= new Heap(10);
}Slide61
61
Priority Queue as Heap
Representation as a Heap
public class
HeapPQ
<T> {
Heap<T>
pq
;
/** Creates a new instance of HeapPQ */
public HeapPQ
() { pq
= new Heap(10);
}Create new heap (
maxsize=10)Slide62
62
Priority Queue as Heap
public void
enqueue
(T e, Priority
pty
){
HeapElement<T> x = new HeapElement
<T>(e, pty);
pq.SiftUp(x);
}public
T serve(Priority pty){
T e; Priority p; e =
pq.heap[1].get_data();
p = pq.heap[1].
get_priority();
pty.set_value(
p.get_value());
pq.heap[1] = pq.heap
[pq.size];
pq.size--;
pq.SiftDown(1);
return
e;
}Slide63
63
Priority Queue as Heap
public void
enqueue
(T e, Priority
pty
){
HeapElement<T> x = new HeapElement
<T>(e, pty);
pq.SiftUp(x);
}public
T serve(Priority pty){
T e; Priority p; e =
pq.heap[1].get_data();
p = pq.heap[1].
get_priority();
pty.set_value(
p.get_value());
pq.heap[1] = pq.heap
[pq.size];
pq.size--;
pq.SiftDown(1);
return
e;
}
Create new heap element (data/priority)Slide64
64
Priority Queue as Heap
public void
enqueue
(T e, Priority
pty
){
HeapElement<T> x = new HeapElement
<T>(e, pty);
pq.SiftUp(x);
}public
T serve(Priority pty){
T e; Priority p; e =
pq.heap[1].get_data();
p = pq.heap[1].
get_priority();
pty.set_value(
p.get_value());
pq.heap[1] = pq.heap
[pq.size];
pq.size--;
pq.SiftDown(1);
return
e;
}
Insert the element at the end of heap, and
SiftUpSlide65
65
Priority Queue as Heap
public void
enqueue
(T e, Priority
pty
){
HeapElement<T> x = new HeapElement
<T>(e, pty);
pq.SiftUp(x);
}public
T serve(Priority pty){
T e; Priority p; e =
pq.heap[1].get_data();
p = pq.heap[1].
get_priority();
pty.set_value(
p.get_value());
pq.heap[1] = pq.heap
[pq.size];
pq.size--;
pq.SiftDown(1);
return
e;
}
Each
enqueue
is inserted in the heap and Sifted Up.
This ensures the element with the lowest priority (min-heap)
or the highest priority (max-heap) is at the top of the heap (index 1).Slide66
66
Priority Queue as Heap
public void
enqueue
(T e, Priority
pty
){
HeapElement<T> x = new HeapElement
<T>(e, pty);
pq.SiftUp(x);
}public
T serve(Priority pty){
T e; Priority p; e =
pq.heap[1].get_data();
p = pq.heap[1].
get_priority();
pty.set_value(
p.get_value());
pq.heap[1] = pq.heap
[pq.size];
pq.size--;
pq.SiftDown(1);
return
e;
}
Get data of the root of the heap (index 1) and store it in eSlide67
67
Priority Queue as Heap
public void
enqueue
(T e, Priority
pty
){
HeapElement<T> x = new HeapElement
<T>(e, pty);
pq.SiftUp(x);
}public
T serve(Priority pty){
T e; Priority p; e =
pq.heap[1].get_data();
p = pq.heap[1].
get_priority();
pty.set_value(
p.get_value());
pq.heap[1] = pq.heap
[pq.size];
pq.size--;
pq.SiftDown(1);
return
e;
}
Get priority of the root of the heap (index 1)Slide68
68
Priority Queue as Heap
public void
enqueue
(T e, Priority
pty
){
HeapElement<T> x = new HeapElement
<T>(e, pty);
pq.SiftUp(x);
}public
T serve(Priority pty){
T e; Priority p; e =
pq.heap[1].get_data();
p = pq.heap[1].
get_priority();
pty.set_value(
p.get_value());
pq.heap[1] = pq.heap
[pq.size];
pq.size--;
pq.SiftDown(1);
return
e;
}
Set “
pty
” value to the root’s priority valueSlide69
69
Priority Queue as Heap
public void
enqueue
(T e, Priority
pty
){
HeapElement<T> x = new HeapElement
<T>(e, pty);
pq.SiftUp(x);
}public
T serve(Priority pty){
T e; Priority p; e =
pq.heap[1].get_data();
p = pq.heap[1].
get_priority();
pty.set_value(
p.get_value());
pq.heap[1] = pq.heap
[pq.size];
pq.size--;
pq.SiftDown(1);
return
e;
}
Copy the last node into the root (delete root)Slide70
70
Priority Queue as Heap
public void
enqueue
(T e, Priority
pty
){
HeapElement<T> x = new HeapElement
<T>(e, pty);
pq.SiftUp(x);
}public
T serve(Priority pty){
T e; Priority p; e =
pq.heap[1].get_data();
p = pq.heap[1].
get_priority();
pty.set_value(
p.get_value());
pq.heap[1] = pq.heap
[pq.size];
pq.size--;
pq.SiftDown(1);
return
e;
}
Update the size to delete last node (delete duplicate node)Slide71
71
Priority Queue as Heap
public void
enqueue
(T e, Priority
pty
){
HeapElement<T> x = new HeapElement
<T>(e, pty);
pq.SiftUp(x);
}public
T serve(Priority pty){
T e; Priority p; e =
pq.heap[1].get_data();
p = pq.heap[1].
get_priority();
pty.set_value(
p.get_value());
pq.heap[1] = pq.heap
[pq.size];
pq.size--;
pq.SiftDown(1);
return
e;
}
SiftDown
the copied last element from the root
down the tree until it satisfies the heap conditionsSlide72
72
Priority Queue as Heap
public void
enqueue
(T e, Priority
pty
){
HeapElement<T> x = new HeapElement
<T>(e, pty);
pq.SiftUp(x);
}public
T serve(Priority pty){
T e; Priority p; e =
pq.heap[1].get_data();
p = pq.heap[1].
get_priority();
pty.set_value(
p.get_value());
pq.heap[1] = pq.heap
[pq.size];
pq.size--;
pq.SiftDown(1);
return
e;
}
Return e (the deleted root)Slide73
73
Priority Queue as Heap
public
int
length(){
return
pq.size;
}public
boolean full(){
return false;
}Slide74
74
HeapSort
Heap can be used for sorting. Two step process:
Step 1: the data is put in a heap.
Step 2: the data are extracted from the heap in sorted order.
HeapSort based on the idea that heap always has the smallest or largest element at the root.Slide75
75
ADT Heap: Implementation
//This method extracts elements in sorted order from the heap. Heap size becomes 0.
public void
HeapSort
(){
while
(size > 1){
swap(heap, 1, size); size--;
SiftDown(1);
}
//Display the sorted elements. for
(int
i = 1; i <=
maxsize; i
++){ System.out.println
(heap[i].
get_priority().get_value
()); }
}Slide76
76
ADT Heap: Implementation
//This method extracts elements in sorted order from the heap. Heap size becomes 0.
public void
HeapSort
(){
while
(size > 1){
swap(heap, 1, size); size--;
SiftDown(1);
}
//Display the sorted elements. for
(int
i = 1; i <=
maxsize; i
++){ System.out.println
(heap[i].
get_priority().get_value
()); }
}
Loop the number of elements in the arraySlide77
77
ADT Heap: Implementation
//This method extracts elements in sorted order from the heap. Heap size becomes 0.
public void
HeapSort
(){
while
(size > 1){
swap(heap, 1, size); size--;
SiftDown(1);
}
//Display the sorted elements. for
(int
i = 1; i <=
maxsize; i
++){ System.out.println
(heap[i].
get_priority().get_value
()); }
}
Swap first element (root) with last elementIn the heap array Slide78
78
ADT Heap: Implementation
//This method extracts elements in sorted order from the heap. Heap size becomes 0.
public void
HeapSort
(){
while
(size > 1){
swap(heap, 1, size); size--;
SiftDown(1);
}
//Display the sorted elements. for
(int
i = 1; i <=
maxsize; i
++){ System.out.println
(heap[i].
get_priority().get_value
()); }
}
Each time reduce the size(size define last element which will be used with swap/
SiftDown)Slide79
79
ADT Heap: Implementation
//This method extracts elements in sorted order from the heap. Heap size becomes 0.
public void
HeapSort
(){
while
(size > 1){
swap(heap, 1, size); size--;
SiftDown(1);
}
//Display the sorted elements. for
(int
i = 1; i <=
maxsize; i
++){ System.out.println
(heap[i].
get_priority().get_value
()); }
}
SiftDown the swapped element (last element) from the root
down the heap until it satisfies heap conditions.Slide80
80
ADT Heap: Implementation
//This method extracts elements in sorted order from the heap. Heap size becomes 0.
public void
HeapSort
(){
while
(size > 1){
swap(heap, 1, size); size--;
SiftDown(1);
}
//Display the sorted elements. for
(int
i = 1; i <=
maxsize; i
++){ System.out.println
(heap[i].
get_priority().get_value
()); }
}
Repeat until the entire heap array is sortedIf it is min-heap, it will be sorted in descending order
If it is max-heap, it will be sorted in ascending order Slide81
81
ADT Heap: Implementation
//This method extracts elements in sorted order from the heap. Heap size becomes 0.
public void
HeapSort
(){
while
(size > 1){
swap(heap, 1, size); size--;
SiftDown(1);
}
//Display the sorted elements. for
(int
i = 1; i <=
maxsize; i
++){ System.out.println
(heap[i].
get_priority().get_value
()); }
}
Print the sorted array(print priorities values)