Transcript huffmancodes

A Data Compression Algorithm:
Huffman Compression
Gordon College
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• Definition: process of encoding which
uses fewer bits
• Reason: to save valuable resources
such as communication bandwidth or
hard disk space
Compression
Compression Types
1. Lossy
– Loses some information during
compression which means the exact
original can not be recovered (jpeg)
– Normally provides better compression
– Used when loss is acceptable - image,
sound, and video files
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Compression Types
1. Lossless
– exact original can be recovered
– usually exploit statistical redundancy
– Used when loss is not acceptable - data
Basic Term: Compression Ratio - ratio of the number of bits in
original data to the number of bits in compressed data
For example: 3:1 is when the original file was 3000 bytes and the
compression file is now only 1000 bytes.
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Variable-Length Codes
• Recall that ASCII, EBCDIC, and
Unicode use same size data structure
for all characters
• Contrast Morse code
QuickTime™ and a
decompressor
are needed to see this picture.
QuickTime™ and a
decompressor
are needed to see this picture.
– Uses variable-length sequences
• The Huffman Compression is a
variable-length encoding scheme
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Variable-Length Codes
• Each character in such a code
– Has a weight (probability) and a length
• The expected length is the sum of the
products of the weights
Goal and lengths for all
the characters minimize the
expected length
0.2 x 2 + 0.1 x 4 +
0.1 x 4 + 0.15 x 3 + 0.45 x 1 = 2.1
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Huffman Compression
• Uses prefix codes (sequence of optimal
binary codes)
• Uses a greedy algorithm - looks at the
data at hand and makes a decision
based on the data at hand.
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Huffman Compression
Basic algorithm
1. Generates a table that contains the
frequency of each character in a text.
2. Using the frequency table - assign each
character a “bit code” (a sequence of bits
to represent the character)
3. Write the bit code to the file instead of the
character.
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Immediate Decodability
• Definition: When no sequence of bits that
represents a character is a prefix of a longer
sequence for another character
Purpose: Can be decoded without waiting
for remaining bits
• Coding scheme to the right is
not immediately decodable
• However this one is
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Huffman Compression
• Huffman (1951)
• Uses frequencies of symbols in a string
to build a variable rate prefix code.
– Each symbol is mapped to a binary string.
– More frequent symbols have shorter codes.
– No code is a prefix of another.
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Huffman Codes
• We seek codes that are
– Immediately decodable
– Each character has minimal expected code
length
• For a set of n characters { C1 .. Cn }
with weights { w1 .. wn }
– We need an algorithm which generates n bit
strings representing the codes
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Cost of a Huffman Tree
• Let p1, p2, ... , pm be the probabilities for the
symbols a1, a2, ... ,am, respectively.
• Define the cost of the Huffman tree T to be
where ri is the length of the path from the root to ai.
• HC(T) is the expected length of the code of a
symbol coded by the tree T. HC(T) is the bit rate of
the code.
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Example of Cost
• Example: a 1/2, b 1/8, c 1/8, d 1/4
HC(T) = 1 x 1/2 + 3 x 1/8 + 3 x 1/8 + 2 x 1/4 = 1.75
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Huffman Tree
• Input: Probabilities p1, p2, ... , pm for symbols a1,
a2, ... ,am, respectively.
• Output: A tree that minimizes the average
number of bits (bit rate) to code a symbol.
That is, minimizes
where ri is the length of the path from the root
to ai. This is a Huffman tree or Huffman code.
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Recursive Algorithm - Huffman Codes
1. Initialize list of n one-node binary trees containing
a weight for each character
2. Repeat the following n – 1 times:
a. Find two trees T' and T" in list with minimal
weights w' and w"
b. Replace these two trees with a binary tree
whose root is w' + w" and whose
subtrees are T' and T"
and label points to these
subtrees 0 and 1
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Huffman's Algorithm
3. The code for character Ci is the bit string
labeling a path in the final binary tree from
the root to Ci
Given characters
The end
result is
with codes
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Huffman Decoding Algorithm
1. Initialize pointer p to root of Huffman tree
2. While end of message string not reached
repeat the following:
a. Let x be next bit in string
b. if x = 0 set p equal to left child pointer
else
set p to right child pointer
c. If p points to leaf
i. Display character with that leaf
ii. Reset p to root of Huffman tree
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Huffman Decoding Algorithm
• For message string 0101011010
– Using Hoffman Tree and decoding algorithm
Click for answer
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Iterative Huffman Tree
Algorithm
1. Form a node for each symbol ai with weight pi;
2. Insert the nodes in a min priority queue ordered by
probability;
3. While the priority queue has more than one element do
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min1 := delete-min;
min2 := delete-min;
create a new node n;
n.weight := min1.weight + min2.weight;
n.left := min1; also associate this link with bit 0
n.right := min2; also associate this link with bit 1
insert(n)
4. Return the last node in the priority queue.
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Example of Huffman Tree
Algorithm (1)
• P(a) =.4, P(b)=.1, P(c)=.3, P(d)=.1, P(e)=.1
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Example of Huffman Tree
Algorithm (2)
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Example of Huffman Tree
Algorithm (3)
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Example of Huffman Tree
Algorithm (4)
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Huffman Code
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In class example
• I will praise you and I will love you Lord
Index Sym
0 space
1 I
2 L
3 a
4 d
5 e
6 i
7 l
8 n
9 o
10 p
11 r
12 s
13 u
14 v
15 w
16 y
Freq
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In class example
• I will praise you and I will love you Lord
Index Sym
0 space
1 I
2 L
3 a
4 d
5 e
6 i
7 l
8 n
9 o
10 p
11 r
12 s
13 u
14 v
15 w
16 y
Freq
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Parent Left Right Nbits Bits
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-1 -1
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01
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-1 -1
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11010
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-1 -1
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00010
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-1 -1
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11110
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-1 -1
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11101
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-1 -1
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0000
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-1 -1
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1100
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-1 -1
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101
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-1 -1
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00011
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-1 -1
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001
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-1 -1
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100110
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-1 -1
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11011
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-1 -1
6
100111
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-1 -1
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1000
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-1 -1
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10010
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-1 -1
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11111
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-1 -1
5
11100
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