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What is Huffman Tree Algorithm?


A weighted binary tree (Minimum weighted binary tree) according to Huffman's algorithm


Huffman's algorithm'

Assume,create a minimum weighted binary tree withn weights i.e W1,W2,...,Wn
Weights are sorted in ascending order.
Getsub tree with two minimum weights as the weights ofexternal nodes.
Include path_lengthof the subtree so obtained into list of the  weights.
Similarly repeated all the process until list of the sequence contains single weight.


From each node symbol = ‘a_i’ with each weight symbol = ‘p_i’ ;
nodes are inseted in a min_priority queue that is ordered by probability;
Whereas, priority_queue has more than 1 element; do
Min-1 = remove – minimum;
Min-2 = remove – minimum;
Now, new node are created i.e node =p
( p.weight) =  (min-1.weight) + (min-2.weight);
(p.left) = min-1;
(p.right) = min-2;
insert( p )
Return the last node in the priority queue.


Huffman tree example:


1) Internal node contain sum of its children’s frequencies

2) Edge to left child is a ‘0’ bit and to a right child is a ‘1’

3) Leaves contain original letters and their frequencies


CODE
a
1 1
b
0 1 1 1
c
0 0
d
0 1 0
e
1 0
f
0 1 1 0


Huffman encoding


Source text: adadadfsadaafeedeaeaeddhhaabaaabbabbhbgccaacc

Frequencies: fre( a ) = 16, Fre( b ) = 6, Fre( c ) = 4, Fre( d ) = 8, Fre( e ) = 5, Fre( f ) = 2, Fre( g ) = 1, Fre( h ) = 3

Tree constructed by repeatedly connecting two lightest nodes

Cost with trivial encoding: (45*3) = 135

Encoded text: 10001000100011100001000101011100011011...010


Optimal Huffman tree


Input: take the input probabilities is , , …,  and for probabilities symbols is , , …, , respectively.

Output: A Huffman tree that minimizes the average number of bits to code a symbol. That is, 

Optimal Huffman treeWhere  ri‘ is the length of the path from the root to ‘ ai .’


Cost of a Huffman Tree


1) Take the input probabilities is , , …,  and for probabilities symbols is , , …, , respectively.

2) Cost of the Huffman tree T is define as 

Cost of the Huffman tree T Where ‘’ is the length of the path from the root to ‘’.

3) The expected length of the code is C(T) and symbol coded by the tree T


Application of Huffman tree: 


1) To obtain an optimal set of codes for symbols

2) Which constitute messages. Each code is a binary string(combinations of 0's and 1's) which will be used for transmission of messages.

3) Fixed length coding

4) Variable length coding




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