Transcript Huffman Tress & Codes

9.4 Huffman
Trees
Michael Alves, Patrick Dugan,
Robert Daniels, Carlos Vicuna
Encoding
• In computer technology, encoding is the process of
putting a sequence of characters into a special
format for transmission or storage purposes.
• Is the term used to reference to the processes
of analog-to-digital conversion, and can be used in
the context of any type of data such as text,
images, audio, video or multimedia.
Huffman Code
Huffman coding is an entropy
encoding algorithm used for lossless data
compression. The term refers to the use of a variablelength code table for encoding a source symbol
(such as a character in a file) where the variablelength code table has been derived in a particular
way based on the estimated probability of
occurrence for each possible value of the source
symbol.
Huffman Code
Huffman coding uses a specific method for
choosing the representation for each symbol,
resulting in a prefix code that expresses the
most common source symbols using shorter
strings of bits than are used for less common
source symbols. Huffman was able to design
the most efficient compression method of this
type: no other mapping of individual source
symbols to unique strings of bits will produce a
smaller average output size when the actual
symbol frequencies agree with those used to
create the code.
Huffman Code
• The running time of Huffman's method is fairly
efficient, it takes O(n log n) operations to construct
it. A method was later found to design a Huffman
code in linear time if input probabilities (also known
as weights) are sorted
Huffman Trees
• Initialize n single-node trees labeled with the
symbols from the given alphabet. Record the
frequency of each symbol in it’s trees root to
indicate the tree’s weight.
• Repeat these steps until a single tree is obtained.
Find two trees with the smallest weight. Make them
the left and right subtree of a new tree and record
the sum of their weights in the root of the new tree
as its weight.
Huffman Trees
• Example: Construct a Huffman tree from the
following data
Symbol
A
Frequency 0.4
B
C
D
_
0.1
0.2
0.15
0.15
In order to generate
binary prefix-free codes
for each symbol in our
alphabet, we label each
left edge of our tree with
0 and every right edge
with 1.
1.0
0
1
0.6
0
1
0.25
B
0.1
0.35
D
0
0.15
1
_
C
0.15
0.20
A
1 0.4
Huffman Trees
• Resulting codewords:
Symbol
A
B
C
D
_
Frequency 0.4
0.1
0.2
0.15
0.15
Codeword 0
100
111
101
110
Example : CAB_ is encoded as 1110100110
Average number of bits per symbol = 1 ∙ 0.4 + 3 ∙ 0.1 + 3 ∙ 0.2 + 3 ∙ 0.15 = 1.75
Compression ratio = (3 – 1.75)/3 * 100 = 42% less memory used than fixedlength encoding
Pseudocode
HuffmanTree(B[0..n − 1])
Constructs Huffman’s tree
Input : An array B[0..n − 1] of weights
Output : A Huffman tree with the given weights assigned to its leaves
initialize priority queue S of size n with single-node trees and priorities equal
to the elements of B[0..n − 1]
while S has more than one element do
Tl ← the smallest-weight tree in S
delete the smallest-weight tree in S
Tr ← the smallest-weight tree in S
delete the smallest-weight tree in S
create a new tree T with Tl and Tr as its left and right subtrees and
the
weight equal to the sum of Tl and Tr weights
insert T into S
return T
Huffman Encoding
Huffman encoding provides the optimal encoding for
all codes using individual letters and those
corresponding frequencies.
Algorithms that use more then that can lead to better
encoding but require more analyzing of the file.
The idea is that if we have letters that are more
frequent than others we represent those with less bits.
Compression Ratio
The compression ration is a standard used to compare
other ways of coding:
fixed bit length - coded bit length
fixed bit length
* 100
This will give the percent of memory used on the encoding
compared to fixed encoding(same length code strings)
If we wanted to compare to different algorithms would
substitute their average bit length instead of fixed
Compression Ratio cont.
Event Name
Probability
Code
Length
A
0.3
00
2
B
0.3
01
2
C
0.13
100
3
D
0.12
101
3
E
0.1
110
3
F
0.05
111
3
Huffman avg bits = (.3 * 2) + (.3 * 2) + (.13 * 3) + (.12 * 3) + (.1 * 3) + (.05 * 3) = 2.4
Compression Ratio = ((3 - 2.4) / 3) * 100 = 20%
This Huffman encoding uses 20% less memory than it’s fixed length
implementation.
Extensive testing with Huffman have shown that it typically falls between 20%-80%
better depending on text.
Real Life Application
• Huffman trees aren’t just used in encoding- they
can be used in any sort of problem involving “yes or
no” decision making.
• “Yes or no” refers to asking multiple questions with
only 2 possible answers (i.e. true or false, heads or
tails, etc.)
• By breaking the problem down into a series of yes or
no questions, you can build a binary tree out of the
possible outcomes. This is called a Decision Tree.
But What Will A Huffman
Tree Do?
• Huffman’s algorithm is designed to create a minimal
length path for a tree with a given weighted path
length.
 This means that the binary tree created will have the shortest paths
possible.
Root
Root
Parent
Leaf
Parent
VS.
Leaf
To get to a given leaf, the
average path length is 2
Leaf
Leaf
To get to a given leaf, the
average path length is 1.5
So…?
• A Huffman Tree is essentially an optimal binary tree
that gives frequently accessed nodes shorter paths
and less frequently accessed nodes longer paths.
• When applied to a Decision Tree, this means that
we will reach the solution, on average, with fewer
“questions”. Hence, we solve the problem faster.
Let’s Try it!
• Consider the problem of guessing which cup a
marble is under. There are 4 cups (c1, c2, c3, and
c4), and you do not get to see which cup the
marble is placed under.
• One decision tree could be:
Is it c1?
c1
Is it c2?
Is it c3?
Average length:
(.25)(1) + (.25)(2) + (.25)(3)(2)
= 2.25
c4
c2
c3
Guide: No <- Question -> Yes
Let’s Try It!
• But what if the person, who cannot be truly random,
is more likely to put it under c4? Assume that c4 now
has a 40% chance, rather than 25%. The other cups
now have a 20% chance.
Is it c4?
0.4
c4
Is it under c3?
Why is this better than the other
Tree, which has an average of 2, rather
Than this tree’s seemingly 2.25?
0.2
Is it c2?
0.2
c1
c3
0.2
c2
Because of c4’s weight, we are more
likely to pick c4. Therefore, we save
Time and the average weight is
(.4)(1) + (.2)(2) + (.4)(3) = 2