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What is Huffman Tree Algorithm?
A weighted binary tree (Minimum weighted binary tree) according to Huffman's algorithm
Assume,create a minimum weighted binary tree with‘n’ 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
|b||0 1 1 1|
|d||0 1 0|
|f||0 1 1 0|
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,
Where ‘ 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
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