Data Structure and Algorithms Huffman Coding Algorithm

Encoding and Compression of Data
Fax Machines
ASCII
Variations on ASCII
min number of bits needed
cost of savings
patterns
modifications

Purpose of Huffman Coding
Proposed by Dr. David A. Huffman in 1952
“A Method for the Construction of Minimum
Redundancy Codes”
Applicable to many forms of data transmission
Our example: text files

The Basic Algorithm
Huffman coding is a form of statistical coding
Not all characters occur with the same frequency!
Yet all characters are allocated the same amount of
space
1 char = 1 byte, be it e or x
Any savings in tailoring codes to frequency of character?
Code word lengths are no longer fixed like ASCII.
Code word lengths vary and will be shorter for the more
frequently used characters.

The (Real) Basic Algorithm
1. Scan text to be compressed and tally occurrence of
all characters.
2. Sort or prioritize characters based on number of
occurrences in text.
3. Build Huffman code tree based on prioritized list.
4. Perform a traversal of tree to determine all code
words.
5. Scan text again and create new file using the
Huffman codes.

Building a Tree
Scan the original text
Consider the following short text:
Eerie eyes seen near lake.
Count up the occurrences of all characters in the text

Building a Tree
What characters are present?
E e r i space
y s n a r l k .

Building a Tree
What is the frequency of each character in the text?
Char Freq. Char Freq. Char Freq.
E 1 y 1 k 1
e 8 s 2 . 1
r 2 n 2
i 1 a 2
space 4 l 1

Building a Tree
Prioritize characters
Create binary tree nodes with character and
frequency of each character
Place nodes in a priority queue
The lower the occurrence, the higher the priority in
the queue
•CS 102

Building a Tree
Prioritize characters
Uses binary tree nodes
public class HuffNode
{
public char myChar;
public int myFrequency;
public HuffNode myLeft, myRight;
}
priorityQueue myQueue;

Building a Tree
The queue after inserting all nodes
Null Pointers are not shown
•CS 102
E
1
i
1
y
1
l
1
k
1
.
1
r
2
s
2
n
2
a
2
sp
4
e
8

Building a Tree
While priority queue contains two or more nodes
Create new node
Dequeue node and make it left subtree
Dequeue next node and make it right subtree
Frequency of new node equals sum of frequency of
left and right children
Enqueue new node back into queue

Building a Tree
E
1
i
1
y
1
l
1
k
1
.
1
r
2
s
2
n
2
a
2
sp
4
e
8

E i
y
1
l
1
k
1
.
1
r
2
s
2
n
2
a
2
sp
4
e
8
2

E i
y
1
l
1
k
1
.
1
r
2
s
2
n
2
a
2
sp
4
e
8

E i
k
1
.
1
r
2
s
2
n
2
a
2
sp
4
e
8
y l

E i
k
1
.
1
r
2
s
2
n
2
a
2
sp
4
e
8
2
y l
2

E i
r
2
s
2
n
2
a
2
sp
4
e
8
2
y l
2
k .
2

E i
n
2
a
2
sp
4
e
8
2
y l
2
k .
2
r s
4

E i
sp
4
e
8
2
y l
2
k .
2
r s
4
n a
4

E i
sp
4
e
8
2
y l
2
k .
2
r s
4
n a
4
4

E i
sp
4
e
82
y l
2
k .
2
r s
4
n a
4 4

E i
sp
e
82
y l
2
k .
2
r s
4
n a
4 4
6

E i
sp
e
8
2
y l
2
k .
2
r s
4
n a
4 4 6
What is happening to the characters
with a low number of occurrences?

E i
sp
e
82
y l
2
k .
2
r s
4
n a
4
4
6
8

E i
sp
e
82
y l
2
k .
2
r s
4
n a
4
4
6 8

E i
sp
e
8
2
y l
2
k .
2
r s
4
n a
4
4
6
8
10

E i
sp
e
8
2
y l
2
k .
2r s
4
n a
4 4
6
8 10

E i
sp
e2
y l
2
k .
2
r s
4
n a
4
4
6
8
10
16

E i
sp
e
2
y l
2
k .
2
r s
4
n a
4
4
6
8
10 16

E i
sp
e
2
y l
2
k .
2
r s
4
n a
4
4
6
8
10
16
26

E i
sp
e
2
y l
2
k .
2
r s
4
n a
4
4
6
8
10
16
26
•After
enqueueing
this node
there is only
one node left
in priority
queue.

Dequeue the single node
left in the queue.
This tree contains the
new code words for each
character.
Frequency of root node
should equal number of
characters in text.
E i
sp
e
2
y l
2
k .
2
r s
4
n a
4
4
6 8
10
16
26
Eerie eyes seen near lake. 26 characters

Encoding the File
Traverse Tree for Codes
Perform a traversal of the
tree to obtain new code
words
Going left is a 0 going right
is a 1
code word is only
completed when a leaf
node is reached
E i
sp
e
2
y l
2
k .
2
r s
4
n a
4
4
6 8
10
16
26

Encoding the File
Traverse Tree for Codes
Char Code
E 0000
i 0001
y 0010
l 0011
k 0100
. 0101
space 011
e 10
r 1100
s 1101
n 1110
a 1111
E i
sp
e
2
y l
2
k .
2
r s
4
n a
4
4
6 8
10
16
26

Encoding the File
Rescan text and encode file
using new code words
Char Code
E 0000
i 0001
y 0010
l 0011
k 0100
. 0101
space 011
e 10
r 1100
s 1101
n 1110
a 1111
0000101100000110011
1000101011011010011
1110101111110001100
1111110100100101
• Why is there no need
for a separator
character?

Encoding the File
Results
Have we made things any
better?
73 bits to encode the text
ASCII would take 8 * 26 =
208 bits
0000101100000110011
1000101011011010011
1110101111110001100
1111110100100101
If modified code used 4 bits per
character are needed. Total bits
4 * 26 = 104. Savings not as great.

Decoding the File
How does receiver know what the codes are?
Tree constructed for each text file.
Considers frequency for each file
Big hit on compression, especially for smaller files
Tree predetermined
based on statistical analysis of text files or file types
Data transmission is bit based versus byte based

Decoding the File
Once receiver has tree it
scans incoming bit stream
0 ⇒ go left
1 ⇒ go right
E i
sp
e
2
y l
2
k .
2
r s
4
n a
4
4
6 8
10
16
26
101000110111101111
01111110000110101
0000101100000110011
1000101011011010011
1110101111110001100
1111110100100101

Summary
Huffman coding is a technique used to compress
files for transmission
Uses statistical coding
more frequently used symbols have shorter code words
Works well for text and fax transmissions
An application that uses several data structures

Data Structure and Algorithms Huffman Coding Algorithm

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