Lecture 21 Heaps and Priority Queues

1
Lecture 21 Heaps and Priority Queues

• May 8, 2008
• Prof. Kyu Ho Park

2
Heaps
? A heap is a binary tree with the following
properties ? It is complete. ? It is
implemented as an array. ? Each node in a heap
satisfies the heap condition , which states that
every nodes key is larger than ( ,or equal
to ) the keys of its children.
88
72
55
70
44
30
50
50
66
22
33
25
3
Complete Binary Tree(CBT)
?Bounds on the size of a CBT ?Height of a
CBT ?Examples of CBT
? There is a natural mapping of each complete
tree into an array of the same size.
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Properties of Heaps

• ? Binary search trees are ordered from left to
  right.
• ? Heaps are binary trees that are ordered from
  bottom to top, so that a traversal along any
  leaf-to-root path will visit the keys
• in ascending order max heaps.
• in descending order min heaps.
• ? Heaps are (1) to implement priority queues,
• (2) to implement the Heap sort
  algorithm.
• ? The parent of node number i is numbered
  (i-1)/2 and its two children are numbered ( 2i1
  ) and ( 2i 2 ).
• ? A heap is a complete binary tree in which the
  keys along leaf-to-root path are ascending. It
  makes a heap partially ordered by the relation
  on its keys.

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Heapify Algorithms

• Input a node x in a complete binary tree.
• Precondition the two subtrees of x are heaps.
• Postcondition the subtree rooted x is a heap.
• 1. Let temp x.
• 2. While x is not a leaf, do steps 3-4.
• 3. Let y be the larger child of x.
• 4. If y.key gt x.key, do steps 5-6.
• 5. Copy y into x.
• 6. Set x y.
• 7. Copy temp into x.

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Heapify Operation
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Build_Heap Algorithm

• Input a complete binary tree.
• Postcondition T is a heap.
• 1. If T is a singleton, return.
• 2. Apply Build_Heap to the left subtree.
• 3. Apply Build_Heap to the right subtree.
• 4. Apply Heapify to the root.

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Heapify Operations
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Priority Queues

• ? A priority queue is a queue whose remove
  operation depends upon the priority ranking of
  its elements.
• ? Ordinary queue FIFO, a Priority queue
  Best-In, First-Out(BIFO).
• ? ADT Priority Queue
• Operations
• 1. Add Insert a given element into the queue.
• 2. Best If the queue is not empty, return the
  element that has the highest priority.
• 3. RemoveBest If the queue is not empty, delete
  and return the element that has the highest
  priority.
• 4. Size Return the number of elements in the
  queue.

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Priority Queues

• ? A prioity queue is a queue whose remove
  operation depends upon the priority rankings of
  its elements. It assumes that the elements can be
  compared to determine which have higher priority.
• ? A PriorityQueue Interface
• 1 public interface PriorityQueue
• 2 public void add(Object object)
• 3 // POSTCONDITION the given object is in this
  queue
• 5 public Object best()
• 6 // RETURN the highest priority element in
  this queue
• 7 // PRECONDITION this queue is not empty
• 9 public Object removeBest()
• 10 // RETURN the highest priority element in
  this queue
• 11 // PRECONDITION this queue is not empty
• 12 // POSTCONDITION the returned object is not
  in this queue
• 14 public int size()
• 15 // RETURN the number of elements in this
  queue
• 16

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Inserting into the priority queue
the Highest Priority Element
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A Huffman Code
ACCEDE
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Huffman Codes

• ? The Huffman tree is a priority queue in which
  the priority of each leaf is the relative
  frequency of the character that it represents,
  and the priority of each internal node is the sum
  of the priorities of its children.
• ? Algorithm Generating a Huffman Code
• Input a sequence of characters.
• Output a bit code for the input characters.
• Postconditions the bit code has the unique
  property and is optimal.
• 1. Tally the frequencies of the input
  characters.
• 2. Load the letter-frequency pairs into a min
  priority queue.
• 3. Coalesce the pairs into a Huffman tree.
• 4. Encode the character at each leaf with the
  bit sequence along its root-to-leaf path.

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Huffman Algorithm
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Coalescing a Huffman Tree

• ? Algorithm Coalescing a Huffman Tree
• Input a min heap Q of integers.
• Output a Huffman tree H of integers.
• Postconditions the elements of Q are the leaves
  of H.
• 1. Restructure Q by interpreting each element as
  itself a singleton tree.
• 2. Repeat steps 3-5 while Q has more than one
  element.
• 3. Remove the two highest-priority trees x and y
  from Q.
• 4. Form the Huffman tree z with children x and
  y.
• 5. Return the remaining element of Q.

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Huffman Tree Properties

• ? A Huffman tree is a binary tree of integers
  with two properties
• 1. Each internal node is the sum of its
  children.
• 2. Its weighted external path length is minimal.

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Huffman Heaps with Minimal Weighted External Path
Length
18223171
18133162

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131629
221638
22
WEPL 2(71) 3(29) 229
WEPL 2(62) 3(38) 238
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Coalescing a Priority Forest of Huffman Trees
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Huffman Codes Another Example

• ? We want to sends a message SUSIE SAYS IT IS
  EASY.
• Frequency Table

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Huffman Code-I

• ? Growing the Huffman tree
• 1.
• 2.
• 3.

Lf(1)
T(1)
U(1)
Y(2)
E(2)
A(2)
I(3)
S(6)
sp(4)
2
T(1)
Y(2)
E(2)
A(2)
I(3)
sp(4)
S(6)
U(1)
Lf(1)
3
2
T(1)
U(1)
Lf(1)
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Huffman Code-I

• ? Growing the Huffman tree
• 4.
• 5.
• 6.

Y(2)
E(2)
A(2)
I(3)
S(6)
sp(4)
3
S(6)
4
A(2)
I(3)
sp(4)
3
Y(2)
E(2)
5
I(3)
4
sp(4)
S(6)
3
A(2)
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Huffman Code-I

• ? Growing the Huffman tree
• 7. 9.
• 8. 10.

S(6)
sp(4)
5
7
9
13
I(3)
4
S(6)
7
22
S(6)
9
7
13
sp(4)
5
9
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Huffman Code-II

• ? Growing the Huffman tree
• 1.
• 2.Min Heap

Lf(1)
T(1)
U(1)
Y(2)
E(2)
A(2)
I(3)
S(6)
sp(4)
Lf(1)
U(1)
T(1)
Y(2)
E(2)
A(2)
I(3)
sp(4)
S(6)
24
Huffman Code-II

• ? Growing the Huffman tree

Lf(1)
T(1)
U(1)
Y(2)
E(2)
A(2)
I(3)
S(6)
sp(4)
2
Lf(1)
Lf(1)
U(1)
U(1)
T(1)
Y(2)
E(2)
A(2)
I(3)
sp(4)
S(6)
25
Huffman Code-II

• ? Growing the Huffman tree

Lf(1)
T(1)
U(1)
Y(2)
E(2)
A(2)
I(3)
S(6)
sp(4)
2
T(1)
Lf(1)
U(1)
2
Y(2)
E(2)
A(2)
I(3)
sp(4)
3
S(6)
T(1)
2
26
Huffman Code-II

• ? Growing the Huffman tree

Lf(1)
T(1)
U(1)
Y(2)
E(2)
A(2)
I(3)
S(6)
sp(4)
2
Y(2)
Lf(1)
U(1)
E(2)
A(2)
4
3
I(3)
sp(4)
S(6)
3
E(2)
Y(2)
T(1)
2
27
Huffman Code-II

• ? Growing the Huffman tree

Lf(1)
T(1)
U(1)
Y(2)
E(2)
A(2)
I(3)
S(6)
sp(4)
5
2
A(2)
3
A(2)
Lf(1)
U(1)
3
I(3)
4
4
sp(4)
S(6)
3
E(2)
Y(2)
T(1)
2
28
Huffman Code-II

• ? Growing the Huffman tree

Lf(1)
T(1)
U(1)
Y(2)
E(2)
A(2)
I(3)
S(6)
sp(4)
5
2
A(2)
3
I(3)
Lf(1)
U(1)
4
sp(4)
4
5
S(6)
3
E(2)
Y(2)
7
T(1)
2
I(3)
4
29
Huffman Code-II

• ? Growing the Huffman tree

Lf(1)
T(1)
U(1)
Y(2)
E(2)
A(2)
I(3)
S(6)
sp(4)
5
2
A(2)
3
sp(4)
Lf(1)
U(1)
5
S(6)
4
7
3
E(2)
Y(2)
7
9
T(1)
2
I(3)
4
sp(4)
5
30
Huffman Code-II

• ? Growing the Huffman tree

Lf(1)
T(1)
U(1)
Y(2)
E(2)
A(2)
I(3)
S(6)
sp(4)
5
2
S(6)
13
A(2)
3
7
Lf(1)
U(1)
S(6)
7
4
9
3
E(2)
Y(2)
7
9
T(1)
2
I(3)
4
sp(4)
5
31
Huffman Tree
22
9
13
sp
5
7
S
4
A
3
I
2
Y
E
T
Lf
U
32
Different Heaps constructed with the same elements
33
Enqueuing
34
Dequeuing
35
Top-down Method
36
Bottom-up Method