Embed Size (px)
Citation preview
9/11/2017
1
Graphs, Trees, Heaps & Priority Queues
Graphs and Trees
Heaps and Heap Sort
Priority Queues using Heaps
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
1
Graphs
Directed graph (or digraph)
G = (V, E)
V: Set of vertex (node)
E: Set of edges (ordered vertex pair)
Undirected graph
G = (V, E)
V: Set of vertex
E: Set of edges (unordered vertex pair)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
2
Graphs
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
3
A
C
B
D
Directed graph (aka, Digraph) G = (V, E)
V = {A, B, C, D, E, F}
E = {(A, B), (C, A), (B, D), (B, C), (C, D), (D, C), (B, B) (F, E)}
E
F
Graphs
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
4
A
C
B
D
E
F
Undirected Graph G = (V, E)
V = {A, B, C, D, E, F}
E = {(A, B), (A, C), (B, D), (B, C), (E, F)}
Trees
A undirected graph is connected if every pair of vertices is connected by a path.
Free Tree – is a connected acyclic, undirected graph.
Forest – a acyclic but possibly disconnected undirected graph
Rooted tree – a free tree where one of the vertices is distinguished from the other. The distinguished vertex is called the root.
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
5
Trees
6
Free Tree Forest
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
9/11/2017
2
Trees If the last edge on the path from
the root R of a tree to a node is (y, x) then y is the parent of x and x is the child of y.
If two nodes have same parent, they are siblings.
A node that has no children is a leaf node.
A non-leaf node is an internallnode.
The length of the path from the root R to a node x is the depth of x in the tree.
The largest depth of any node in tree T is the height of tree T.
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
7
R
A C DB
GF
H
E
Binary Trees
A binary tree is a tree data structure in which each node has at most two children.
Typically the child nodes are called leftchild and right child.
Binary trees are commonly used to implement binary search trees and
heaps.
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
8
Binary Trees
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
9
Height-Balanced Binary Tree???
Binary Trees
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
10
Height-Balanced Binary Tree!
An empty tree is height-balanced.A non-empty binary tree T is balanced if:1. Left subtree of T is balanced2. Right subtree of T is balanced3. The difference between heights of left subtree
and right subtree is not more than 1.
Binary Tree
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
11
Unbalanced Binary Tree
Sorting with Recursion(Heap Sort: Heap)
The (binary) heap data structure is an arraythat can be viewed as a complete binary tree.
Complete binary tree: A tree that is completely filled on all levels except possibly the lowest, where all the nodes must be as far left as possible.
Heap property: the key of a parent is always greater than or equal to the keys of its children for a max-heap (less than or equal to the keys of its children for a min-heap).
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
12
9/11/2017
3
Sorting with Recursion(Heap Sort: Heap)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
13
16
14 10
8 7 9 3
2 4 1
16 14 10 8 7 9 3 2 4 1
0 1 2 3 4 5 6 7 8 9
Heap size = 10
Sorting with Recursion(Heap Sort: Heap)
The root of the tree is A[0], and given the index i of a node, the indices of its parent, left child and right child can be computed simply:
Parent (i)
If i is even
Return ((i - 1)/2)
Else
Return (i / 2)
LeftChild (i)
Return (2i + 1)
RightChild (i)
Return (2i + 2)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
14
16
14 10
8 7 9 3
2 4 1
Sorting with Recursion(Heap Sort: Heap Functions)
Since a heap of n elements is based on a complete binary tree, the height is (log2 n).
There are five basic algorithms regarding heap.
1. HEAPIFY: is the key to maintain the heap property.
2. BUILD-HEAP: produces a heap from an unordered input array.
3. HEAPSORT: sorts an array in place.
4. EXTRACT-MAX: extracts the maximum value from the heap
5. INSERT: allows the heap data structure to be used as a priority queue.
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
15
Sorting with Recursion(Heap Sort: Heap: HEAPIFY)
The HEAPIFY function is an important procedure for manipulating heaps.
Its inputs are an array A and an index i into the array.
When HEAPIFY is called, it is assumed that the binary trees rooted at LeftChild(i) and RightChild(i) are heaps, but that A[i] may be smaller than its children, thus violating the heap property.
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
16
Sorting with Recursion(Heap Sort: Heap: HEAPIFY)
HEAPIFY(A, i) {
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L < Heapsize(A) and A[L] > A[i]
4 Largest = L;
5 else
6 Largest = i;
7 if R < Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } // end if
} // end of HEAPIFY
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
17
Sorting with Recursion(Heap Sort: Heap: HEAPIFY)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
18
10
14 7 9 3
2 8 1
10 14 7 9 3 2 8 1
HEAPIFY (A, 0)
0 1 2 3 4 5 6 7 8 9
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L < Heapsize(A) and A[L] > A[i]
4 Largest = L;
5 else
6 Largest = i;
7 if R < Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
4
16
4
0
1 2
3 4 5 6
7 8 9
16
9/11/2017
4
Sorting with Recursion(Heap Sort: Heap: HEAPIFY)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
19
16
10
7 9 3
2 8 1
16 10 7 9 3 2 8 1
HEAPIFY (A, 1)
0 1 2 3 4 5 6 7 8 9
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L < Heapsize(A) and A[L] > A[i]
4 Largest = L;
5 else
6 Largest = i;
7 if R < Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY14
4
14
0
1 2
3 4 5 6
7 8 9
4
Sorting with Recursion(Heap Sort: Heap: HEAPIFY)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
20
16
14 10
7 9 3
2 1
16 14 10 7 9 3 2 1
HEAPIFY (A, 3)
0 1 2 3 4 5 6 7 8 9
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L < Heapsize(A) and A[L] > A[i]
4 Largest = L;
5 else
6 Largest = i;
7 if R < Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
8
4
8
0
1 2
3 4 5 6
7 8 9
4
Sorting with Recursion(Heap Sort: Heap: HEAPIFY)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
21
16
14 10
8 7 9 3
2 4 1
16 14 10 8 7 9 3 2 4 1
0 1 2 3 4 5 6 7 8 9
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L < Heapsize(A) and A[L] > A[i]
4 Largest = L;
5 else
6 Largest = i;
7 if R < Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
0
1 2
3 4 5 6
7 8 9
Sorting with Recursion(Heap Sort: Heap: Run Time for HEAPIFY)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
22
HEAPIFY(A, i){
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L < Heapsize(A) and A[L] > A[i]
4 Largest = L;
5 else
6 Largest = i;
7 if R < Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } // end if
} // end of HEAPIFY
(1): constant time
Worst Case(log2n): Height of tree
Sorting with Recursion(Heap Sort: Heap: Build_Heap)
We can use the procedure HEAPIFY in a bottom-up manner to convert an array A[0..n-1] into a heap.
Since elements in the sub-array A[(n/2)..n-1] are all leaves of the tree, The procedure Build-Heap goes through the internal nodes (remaining nodes) of the tree ( A[0..(n/2)-1] ) and runs HEAPIFY on each node.
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
23
Sorting with Recursion(Heap Sort: Heap: Build_Heap)
Build_Heap (A)
{
1 heap_size = |A|
2 For i = ((|A|/2) - 1) downto 0
3 HEAPIFY (A, i)
}
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
24
9/11/2017
5
Sorting with Recursion(Heap Sort: Heap: Build_Heap)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
25
4
1 3
2 16 9 10
14 8 7
4 1 3 2 16 9 10 14 8 7
0 1 2 3 4 5 6 7 8 9
Build_Heap (A)
{
1 heap_size = |A|
2 For i = ((|A|/2) - 1) down to 0
3 HEAPIFY (A, i)
}
heap_size = 10i = 10/2 – 1 = 4
0
1 2
3 4 5 6
7 8 9
HEAPIFY(A, 4)
Sorting with Recursion(Heap Sort: Heap: Build_Heap)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
26
Build_Heap (A)
{
1 heap_size = |A|
2 For i = ((|A|/2) - 1) down to 0
3 HEAPIFY (A, i)
}
4
1 3
16 9 10
8 7
4 1 3 16 9 10 8 7
0 1 2 3 4 5 6 7 8 9
0
1 2
3 4 5 6
7 8 9
heap_size = 10i = i - 1 = 4 -1=3HEAPIFY(A, 3)
2
14
2 14
Sorting with Recursion(Heap Sort: Heap: Build_Heap)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
27
4
1
14 16 9
2 8 7
4 1 14 16 9 2 8 7
0 1 2 3 4 5 6 7 8 9
Build_Heap (A)
{
1 heap_size = |A|
2 For i = ((|A|/2) - 1) down to 0
3 HEAPIFY (A, i)
}
heap_size = 10i = 3 – 1 = 2
0
1 2
3 4 5 6
7 8 9
HEAPIFY(A, 2)
3
10
3 10
Sorting with Recursion(Heap Sort: Heap: Build_Heap)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
28
4
10
14 9 3
2 8 7
4 10 14 9 3 2 8 7
0 1 2 3 4 5 6 7 8 9
Build_Heap (A)
{
1 heap_size = |A|
2 For i = ((|A|/2) - 1) down to 0
3 HEAPIFY (A, i)
}
heap_size = 10i = i – 1 = 1
0
1 2
3 4 5 6
7 8 9
HEAPIFY(A, 1)
1
16
1 16
Sorting with Recursion(Heap Sort: Heap: Build_Heap)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
29
4
10
14 9 3
2 8
4 10 14 9 3 2 8
0 1 2 3 4 5 6 7 8 9
Build_Heap (A)
{
1 heap_size = |A|
2 For i = ((|A|/2) - 1) down to 0
3 HEAPIFY (A, i)
}
heap_size = 10i = i – 1 = 1
0
1 2
3 4 5 6
7 8 9
HEAPIFY(A, 4)
16
1
16 1
7
7
Sorting with Recursion(Heap Sort: Heap: Build_Heap)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
30
10
14 7 9 3
2 8 1
10 14 7 9 3 2 8 1
0 1 2 3 4 5 6 7 8 9
Build_Heap (A)
{
1 heap_size = |A|
2 For i = ((|A|/2) - 1) down to 0
3 HEAPIFY (A, i)
}
heap_size = 10i = i – 1 = 0
0
1 2
3 4 5 6
7 8 9
HEAPIFY(A, 0)4
16
4 16
9/11/2017
6
Sorting with Recursion(Heap Sort: Heap: Build_Heap)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
31
16
10
7 9 3
2 8 1
16 10 7 9 3 2 8 1
0 1 2 3 4 5 6 7 8 9
Build_Heap (A)
{
1 heap_size = |A|
2 For i = ((|A|/2) - 1) down to 0
3 HEAPIFY (A, i)
}
heap_size = 10
0
1 2
3 4 5 6
7 8 9
HEAPIFY(A, 1)
4
4
14
14
Sorting with Recursion(Heap Sort: Heap: Build_Heap)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
32
16
14 10
7 9 3
2 1
16 14 10 7 9 3 2 1
0 1 2 3 4 5 6 7 8 9
Build_Heap (A)
{
1 heap_size = |A|
2 For i = ((|A|/2) - 1) down to 0
3 HEAPIFY (A, i)
}
heap_size = 10
0
1 2
3 4 5 6
7 8 9
HEAPIFY(A, 3)
4
4
8
8
Sorting with Recursion(Heap Sort)
The Heap Sort algorithm takes advantage of the heap property.
1. The algorithm starts by building a heap out of the array.
2. Since the root of a heap is the largest element in the heap, we exchange it with the last element in the heap. The largest element is now in its correct final position!
3. Heapify sub array A[1 .. n-1]
4. Call Heap sort recursively with sub-heap
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
33
Sorting with Recursion(Heap Sort)
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
34
Θ(𝑛𝑙𝑜𝑔2𝑛)
Θ(𝑙𝑜𝑔2𝑛)
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
35
4
1 3
2 16 9 10
14 8 7
4 1 3 2 16 9 10 14 8 7
0 1 2 3 4 5 6 7 8 9
heap_size = 10 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
36
14 10
8 7 9 3
2 4
14 10 8 7 9 3 2 4
0 1 2 3 4 5 6 7 8 9
heap_size = 10 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
1
16
16 1
9/11/2017
7
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
37
10
8 7 9 3
2 4 16
10 8 7 9 3 2 4 16
0 1 2 3 4 5 6 7 8 9
heap_size = 9 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
HEAPIFY(A, 0)1
14
1 14
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
38
14
10
7 9 3
2 4 16
14 10 7 9 3 2 4 16
0 1 2 3 4 5 6 7 8 9
heap_size = 9 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
HEAPIFY(A, 1)
1
1
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
8
8
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
39
14
8 10
7 9 3
2 16
14 8 10 7 9 3 2 16
0 1 2 3 4 5 6 7 8 9
heap_size = 9 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
HEAPIFY(A, 3)HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
1
1
4
4
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
40
8 10
4 7 9 3
2 16
8 10 4 7 9 3 2 16
0 1 2 3 4 5 6 7 8 9
heap_size = 9 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
14
1
14 1
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
41
8
4 7 9 3
2 14 16
8 4 7 9 3 2 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 8 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
1
1
HEAPIFY(A, 0)
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
10
10
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
42
10
8
4 7 3
2 14 16
10 8 4 7 3 2 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 8 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
HEAPIFY(A, 1)
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
1
1 9
9
9/11/2017
8
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
43
8 9
4 7 1 3
14 16
8 9 4 7 1 3 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 8 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
10
2
10 2
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
44
8
4 7 1 3
10 14 16
8 4 7 1 3 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 7 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
HEAPIFY(A, 0)2
9
2 9
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
45
9
8
4 7 1
10 14 16
9 8 4 7 1 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 7 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
HEAPIFY(A, 2)
2
2
3
3
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
46
8 3
4 7 1
10 14 16
8 3 4 7 1 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 7 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
2
2
9
9
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
47
3
4 7 1 9
10 14 16
3 4 7 1 9 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 6 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
2
2
HEAPIFY(A, 0)
8
8
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
48
8
3
4 1 9
10 14 16
8 3 4 1 9 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 6 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
HEAPIFY(A, 1)
2
2
7
7
9/11/2017
9
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
49
7 3
4 2 9
10 14 16
7 3 4 2 9 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 6 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
8
1
8 1
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
50
3
4 2 8 9
10 14 16
3 4 2 8 9 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 5 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
1
1
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
HEAPIFY(A, 0)
7
7
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
51
7
3
2 8 9
10 14 16
7 3 2 8 9 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 5 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
HEAPIFY(A, 1)
1
1
4
4
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
52
3
1 8 9
10 14 16
3 1 8 9 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 5 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
7
4
7 4
2
2
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
53
3
1 7 8 9
10 14 16
3 1 7 8 9 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 4 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
2
4
2 4
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
HEAPIFY(A, 0)
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
54
3
7 8 9
10 14 16
3 7 8 9 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 4 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
4
2
4 2
1
1
9/11/2017
10
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
55
7 8 9
10 14 16
7 8 9 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 3 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
1
2
1 2
4
4
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
HEAPIFY(A, 0)
3
3
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
56
7 8 9
10 14 16
7 8 9 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 3 0
1 2
3 4 5 6
7 8 9
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
2
3 2
4
4
1
1
3
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
57
7 8 9
10 14 16
7 8 9 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 2 0
1 2
3 4 5 6
7 8 9
2
1 2
4
4
3
3
1HEAPIFY(A, 0)
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
58
7 8 9
10 14 16
7 8 9 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 2 0
1 2
3 4 5 6
7 8 9
1
2 1
4
4
3
3
2HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
59
7 8 9
10 14 16
7 8 9 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 1 0
1 2
3 4 5 6
7 8 9
2
1 2
4
4
3
3
1HEAPIFY(A, 0)
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
Sorting with Recursion(Heap Sort)
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
60
7 8 9
10 14 16
7 8 9 10 14 16
0 1 2 3 4 5 6 7 8 9
heap_size = 1 0
1 2
3 4 5 6
7 8 9
2
1 2
4
4
3
3
1HEAPIFY(A, 0)
HEAPSORT(A)
{
1 Build_Heap(A);
2 for i = |A|-1 downto 1
{
3 swap(A[0], A[i])
4 heap_size = heap_size – 1;
5 HEAPIFY(A, 0)
}
}
9/11/2017
11
Priority Queues with Heaps
A priority queue is a data structure for maintaining a set S of elements each with an associated value called key.
We can use a heap to implement a priority queue
A priority queue needs to support the following operations.
1. Insert(S, x) : insert the element x into the set S.
2. Maximum(S): returns the elements of S with the largest key.
3. Extract_Max(S): removes and returns the elements of S with the largest key.
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
61
Priority Queues with Heaps
Applications with priority queues A priority queue can be used to schedule jobs on a shared
computer – each job with priority vector is resided in thepriority queue. A job with highest priority is chosen for nextservice.
A priority queue can also be used in an event-drivensimulator - The item in the priority queue are event to besimulated, each with an associated time of occurrence thatserves as its key. The event must be simulated in order oftheir time of occurrence, since the simulation of an evencan cause other events to be simulated in the future.
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
62
Priority Queues with Heaps (Extract Maximum)
Extract_Max(A)
{
1 if (heap_size < 1)
2 Error “heap underflow”
3 else
{
4 max = A[0];
5 A[0] = A[heap_size - 1];
6 heap_size = heap_size – 1;
7 HEAPIFY (A, 0);
8 return (max);
}
}
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
63
Θ(1)
Θ(𝑙𝑜𝑔2𝑛)
Priority Queues with Heaps (Extract Maximum)
18
16 10
14 7 9 3
2 8 1
18 16 10 14 7 9 3 2 8 1
0 1 2 3 4 5 6 7 8 9 10
4
4
heap_size = 11
max = 18
64COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
0
1 2
3 4 5 6
7 8 9 10
Extract_Max(A)
{
1 if (heap_size < 1)
2 Error “heap underflow”
3 else
{
4 max = A[0];
5 A[0] = A[heap_size -1];
6 heap_size = heap_size – 1;
7 HEAPIFY (A, 0);
8 return (max);
}
}
4
4
Priority Queues with Heaps (Extract Maximum)
10
14 7 9 3
2 8 1
10 14 7 9 3 2 8 1
0 1 2 3 4 5 6 7 8 9 10
4
4
heap_size = 10
max = 18
65COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
0
1 2
3 4 5 6
7 8 9 10
Extract_Max(A)
{
1 if (heap_size < 1)
2 Error “heap underflow”
3 else
{
4 max = A[0];
5 A[0] = A[heap_size -1];
6 heap_size = heap_size – 1;
7 HEAPIFY (A, 0);
8 return (max);
}
}
HEAPIFY(A, 0)4
16
4 16
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
Priority Queues with Heaps (Extract Maximum)
10
7 9 3
2 8 1
10 7 9 3 2 8 1
0 1 2 3 4 5 6 7 8 9 10
heap_size = 10
max = 18
66COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
0
1 2
3 4 5 6
7 8 9
Extract_Max(A)
{
1 if (heap_size < 1)
2 Error “heap underflow”
3 else
{
4 max = A[0];
5 A[0] = A[heap_size -1];
6 heap_size = heap_size – 1;
7 HEAPIFY (A, 0);
8 return (max);
}
}
HEAPIFY(A, 1)
16
4
16 4
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
14
14
9/11/2017
12
Priority Queues with Heaps (Extract Maximum)
10
7 9 3
2 1
10 7 9 3 2 1
0 1 2 3 4 5 6 7 8 9 10
heap_size = 10
max = 18
67COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
0
1 2
3 4 5 6
7 8 9
Extract_Max(A)
{
1 if (heap_size < 1)
2 Error “heap underflow”
3 else
{
4 max = A[0];
5 A[0] = A[heap_size -1];
6 heap_size = heap_size – 1;
7 HEAPIFY (A, 0);
8 return (max);
}
}
HEAPIFY(A, 3)
16
14
16 14
HEAPIFY(A, i)
{
1 L = LeftChild(i);
2 R = RightChild(i);
3 if L Heapsize(A) and A[L] >A[i]
4 Largest = L;
5 else
6 Lagest = I;
7 if R Heapsize(A) and A[R] > A[Largest]
8 Largest = r;
9 if Largest i
10 {
11 Swap (A[i], A[Largest]);
12 HEAPIFY(A, Largest);
13 } //end if
}// end of HEAPIFY
4
4
8
8
Priority Queues with Heaps (Extract Maximum)
10
7 9 3
2 1
10 8 7 9 3 2 4 1
0 1 2 3 4 5 6 7 8 9 10
heap_size = 10
max = 18
68COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
0
1 2
3 4 5 6
7 8 9
Extract_Max(A)
{
1 if (heap_size < 1)
2 Error “heap underflow”
3 else
{
4 max = A[0];
5 A[0] = A[heap_size -1];
6 heap_size = heap_size – 1;
7 HEAPIFY (A, 0);
8 return (max);
}
}
16
14
16 14
8
4
Heap_Insert inserts a node into heap A.
Increase the heap size by one
Insert new element in the correct location in the heap.
Uses comparisons and swaps to determine the correct location for the new element and insert it.
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
69
Priority Queues with Heaps (Insert)
Heap_Insert (A, key)
{
1 heap_size = heap_size + 1;
2 i = heap_size – 1 // highest index in the heap
3 while i > 0 and A[Parent(i)] < key
{
4 A[i] = A[Parent(i)];
5 i = Parent(i);
}
6 A[i] = key // copy the key value in the right location
}
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
70
Θ(1)
Θ(1)
O(𝑙𝑜𝑔2𝑛)
The running time of Heap_Insert on an n-element heap is O(𝑙𝑜𝑔2𝑛)
Priority Queues with Heaps (Insert)
Priority Queues with Heap (Insert)
10
7 9 3
2 1
10 8 7 9 3 2 4 1
0 1 2 3 4 5 6 7 8 9 10
Initial heap_size = 10
71COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
0
1 2
3 4 5 6
7 8 9
16
14
16 14
8
4
Insert _Heap(A, 15)
Heap_Insert (A, key)
{
1 heap_size = heap_size + 1;
2 i = heap_size – 1
3 while i > 0 and A[Parent(i)] < key
{
4 A[i] = A[Parent(i)];
5 i = Parent(i);
}
6 A[i] = key
}
heap_size = 11
10
15
Priority Queues with Heap (Insert)
10
7 9 3
2 1
10 8 7 9 3 2 4 1
0 1 2 3 4 5 6 7 8 9 10
Initial heap_size = 10
72COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
0
1 2
3 4 5 6
7 8 9
16
14
16 14
8
4
Insert _Heap(A, 15)
Heap_Insert (A, key)
{
1 heap_size = heap_size + 1;
2 i = heap_size – 1
3 while i > 0 and A[Parent(i)] < key
{
4 A[i] = A[Parent(i)];
5 i = Parent(i);
}
6 A[i] = key
}
10 i=10
7
7
heap_size = 11
15
9/11/2017
13
Priority Queues with Heap (Insert)
14 10
7 9 3
2 1
14 10 8 7 9 3 2 4 1
0 1 2 3 4 5 6 7 8 9 10
Initial heap_size = 10
73COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
0
1 2
3 4 5 6
7 8 9
16
16
8
4
Insert _Heap(A, 15)
Heap_Insert (A, key)
{
1 heap_size = heap_size + 1;
2 i = heap_size – 1
3 while i > 0 and A[Parent(i)] < key
{
4 A[i] = A[Parent(i)];
5 i = Parent(i);
}
6 A[i] = key
}
7
710
i=4
7
15
heap_size = 11
14
14
15
Priority Queues with Heap (Insert)
15 10
14 9 3
2 1
15 10 8 14 9 3 2 4 1
0 1 2 3 4 5 6 7 8 9 10
74COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
0
1 2
3 4 5 6
7 8 9
16
16
8
4
Heap_Insert (A, key)
{
1 heap_size = heap_size + 1;
2 i = heap_size – 1
3 while i > 0 and A[Parent(i)] < key
{
4 A[i] = A[Parent(i)];
5 i = Parent(i);
}
6 A[i] = key
}
7
710
COSC 320 - Fall 2017
Slide courtesy of Dr. Sang-Eon Park
75
Review: Sorting Algorithms
Algorithm Best Case Worst Case
Insertion Θ(𝑛) Θ(𝑛2)
Selection Θ(𝑛2) Θ(𝑛2)
Bubble Θ(𝑛2) Θ(𝑛2)
Shell Θ 𝑛𝑘
1 < 𝑘 < 2
Θ 𝑛𝑘
1 < 𝑘 < 2
Merge Θ(𝑛𝑙𝑜𝑔2𝑛) Θ(𝑛𝑙𝑜𝑔2𝑛)
Heap Θ(𝑛𝑙𝑜𝑔2𝑛) Θ(𝑛𝑙𝑜𝑔2𝑛)
Quick Θ(𝑛𝑙𝑜𝑔2𝑛) Θ(𝑛2)
