CSC 2300 Data Structures

1
CSC 2300Data Structures Algorithms

• April 27, 2007
• Chap. 10. Algorithm Design Techniques

2
Today

• File Compression
• Huffman Code

3
ASCII

• What does ASCII stand for?
• The ASCII character set consists of about 100
  printable characters.
• How many bits to represent these characters?
• The set includes some nonprintable characters.
• An 8th bit is added as a parity bit.

4
Example

• A file with only the characters a, e, i, s, t,
  blankspace, newline.
• There are seven characters, and so three bits are
  sufficient.
• i see a seat
• 010101011001001101000101011001000100110 (39
  bits)
• How to do better?

5
Binary Tree

• Binary tree
• The data reside only at the leaves.
• Can you improve this representation?

6
Example

• newline becomes 11
• i see a seat
• 01010101100100110100010101100100010011 (38 bits)
• A reduction of 1 bit.
• Want more significant improvement.
• How?

7
The Two Trees

• What can you say about the structure of the
  better tree?
• It a a full tree.
• All nodes either are leaves or have two children.
• An optimal code will always have this property.
• Why?
• Nodes with only one child can always move up one
  level.

8
Prefix Code

• If the characters are placed only at the leaves,
  the given sequence of bits can be decoded
  unambiguously.
• Prefix code no character code is a prefix of
  another character code.
• Example 01001111000010110001000111
• What is it?
• is
• a tie

9
Optimal Prefix Code

• Binary tree
• How to find optimal code?

10
Our Example

• i see a seat
• 1011000000101110011100000010010001 (34 bits)
• The code in the table is not optimal for our
  example.
• Why not?
• Exercise. Find the optimal code for our example.

11
Huffmans Algorithm

• Assume that there are C characters.
• Maintain a forest of trees.
• The weight of a tree is equal to the sum of the
  frequencies of its leaves.
• For C 1 times, select the two trees T1 and T2
  of smallest weights, breaking ties arbitrarily,
  and form a new tree with subtrees T1 and T2.
• At the beginning, there are C single-node trees.
  At the end, there is one single tree, which is
  the optimal Huffman coding tree.

12
Example

• Initial stage
• After first merge

13
Example

• After first merge
• After second merge
• After third merge

14
Example

• After third merge
• After fourth merge

15
Example

• After fourth merge
• After fifth merge

16
Example

• After fifth merge
• After final merge

17
Implementation

• If we maintain the trees in a priority queue,
  ordered by weight, what is the running time?
• O( C log C ).
• We say that Huffmans method is a two-pass
  algorithm. What are the two passes?
• The first pass selects the frequency data and the
  second pass performs the encoding.