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Dec 24, 2014
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Data Compression
Muhammad Raza Master (B12101085)
Muhammad Ali Mehmood (B12101065)
Syed Faraz Naqvi (B12101123)
-Department of Computer Science, University of Karachi
What is data
compression???
Reduction in size of data
Why??
Save storage when saving information
Save time when communicating information
Types of data compression
Compression
Lossless Lossy
Practical
Examples
• Image Compression
• Audio Compression
• Video compression
• All Sort of DataCompression
Tree
TREE
ROOT
Sub Tree/Parent Node
Left child Right child
• Sum of children’s frequency • Reference of B-Tree(0/1)
* Char variable * Frequency * Reference of B-Tree(0/1)
APPLICATION
• Find an object with a certain property in a collection of objects of a certain type
• Items in a list be stored so that an item can be easily located
• Efficient encoding of set of characters by bit strings
TRAVERSING IN TREE
• IN-ORDER TRAVERSAL
• PREORDER TRAVERSAL
• POSTORDER TRAVERSAL
4 12 18 24
10 22
31 44 66 90
35 70
15 50
25
Pre-Order In-Order Post-order1. Visit the root Traverse the left subtree Traverse the left subtree2. Traverse the left subree Visit the root Traverse the right subtree3. Traverse the right subtree Traverse the right subtree Visit the root
Pre-Order: 25, 15, 10, 4, 12, 22, 18, 24, 50, 35, 31, 44, 70, 66, 90In-Order: 4, 10, 12, 15, 18, 22, 24, 25, 31, 35, 44, 50, 66, 70, 90Post Order: 4, 12, 10, 18, 24, 22, 15, 31, 44, 35, 66, 90, 70, 50, 25
Huffman
Encoding
• By Dr. David Huffman (1952)
• First data compression algorithm
• An example of ‘LOSSLESS DATA COMPRESSION’
• Binary tree is used to construct Huffman encoding algorithm
Introduction
Basic Idea
Largest occurring char has the least encoded bit.
Save bits by encoding frequently used characters with fewer bits than rarely used characters
Algorithm
HUFFMAN(X)
• Compute frequency f(c) for each character c in X.• Let Q be an empty priority queue• Insert every character c into Q as singleton trees with
key f(c)• while Q.SIZE() > 1
– Do• f1 ← Q.MIN-KEY()• T1 ← Q.REMOVE-MIN()• f2 ← Q.MIN-KEY()• T2 ← Q.REMOVE-MIN()• Let T be a new tree with left subtree T1 and right subtree T2• Q.INSERT(T, f1 + f2)
• Return Q.REMOVE-MIN()
it was the best of times it was the worst of times.
Symbol Count
LF 1
b 1
r 1
f 2
h 2
m 2
a 2
w 3
o 3
i 4
e 5
s 6
t 8
space 11
(full stop) = LF
Example:
Symbol BitsLF 101010b 101011r 10100f 11000h 11001m 11010a 11011w 0010o 0011i 1011e 000s 100t 111space 01
Demonstration of
Huffman
Encoding
Example#1:
HumeraTariq
Symbol Count
H 1
u 1
m 1
e 1
r 2
a 2
T 1
I 1
q 1
H u m e T i
2 2 2 q
4 3 r a
7 4
110 1
1
1
11
1
1
10
0
0
0
00
0
m = HumeraTariq
Symbol Bits
H 0000
u 0001
m 0010
e 0011
r 10
a 11
T 0100
i 0101
q 0110
Compressed Bit-stream
C(m) = 000000010010001110110100111001010110
Proposition
The length of the encoded bit-stream is the sum over all letters of the number of occurrences times the number of
bits per occurrence
Compressed bit-stream = frequency * Distance
Proof
E.g: m= HumeraTariq• At distance:– 4: six leaf (‘H’, ‘u’, ‘m’, ‘e’, ‘T’, ‘i’, with total
frequency 6) – 3: one leaf (‘q’, with frequency 1)– 2: two leaf nodes (‘r’ and ‘a’, with total frequency
4)
• Compressed bit-stream = frequency * Distance
• total = 4·6 + 3·1 + 2.4 = 35 is the length of compressed bit-stream as expected
Proved!!
Complexity
Let d be the number of symbols, n be the length of the input
Huffman’s algorithm runs in O(n + d log d) time
Success of
Huffman Coding
We can apply it to any bytestream
Milestone of LZW compression
REFERENCES
• Robert Sedgewick and Kevin Wayne - Algorithms, (4th edition)
• https://blog.itu.dk/BADS-F2009/files/2009/04/46-huffman.pdf
• Discrete Mathematics and Its Applications (7th Edition-Rosen)