Functional Programming Lecture 15 - Case Study: Huffman Codes

1
Functional ProgrammingLecture 15 - Case
StudyHuffman Codes
2
The Problem

• Design a coding/decoding scheme and implement in
  Haskell.
• This requires
• - an algorithm to encode a message,
• - an algorithm to decode a message,
• - an implementation.

3
Fixed and Variable Length Codes

• A fixed length code assigns the same number of
  bits to each code word.
• E.g. ASCII letter -gt 7 bits (up to 128 code
  words)
• So to encode the string at we need 14 bits.
• A variable length code assigns a different number
  of bits to each code word, depending on the
  frequency of the code word. Frequent words are
  assigned short codes infrequent words are
  assigned long codes.
• E.g.
• a
  at encoded by 011
• 
  0 for go left
• b t
  1 for go right
• tree to encode and decode
  

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Coding
0 1 a 0
1 b t
a is encoded by 1 bit, 0 b is encoded by 2
bits, 10 t is encoded by 2 bits, 11 An important
property of a Huffman code is that the codes are
prefix codes no code of a letter (code word) is
the prefix of the code of another letter (code
word). E.g. 0 is not a prefix of 10 or 11
10 is not a prefix of 0 or 11 11 is not a
prefix of 0 or 10 So, aa is encoded by 00.
ba is encoded by 100.

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Decoding
0 1 a 0
1 b t
The encoded message 1001111011is decoded
as 10 - b 0 - a 11 - t 11 - t 0 - a 11 -
t In view of the frequency of t, this is
probably not a good code. t should be encoded by
1 bit! ps. Morse code is a type of Huffman
code.

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A Haskell Implementation

• Types
• -- codes --
• data Bit L R deriving (Eq, Show)
• type Hcode Bit
• -- Huffman coding tree --
• -- characters at leaf nodes, plus frequencies --
• -- frequencies as well at internal nodes --
• data Tree Leaf Char Int Node Int Tree Tree
• Assume that codes are kept in table (rather than
  read off a tree).
• -- table of codes --
• type Table (Char, Hcode)

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Encoding

• -- encode a message according to code table --
• -- encode each character and concatenate --
• codeMessage Table -gt Char -gt Hcode
• codeMessage tbl concat . map (lookupTable tbl)
• -- lookup the code for a character in code table
  --
• lookupTable Table -gt Char -gt Hcode
• lookupTable c error lookupTable
• lookupTable ((ch,code)tbl) c
• ch c code
• otherwise lookupTable tbl c

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Decoding

• -- decode a message according to code tree --
  
• -- if at a leaf node, then character is decoded,
  --
• -- start again at root
  --
• -- if at an internal node, then follow sub-tree
  --
• -- according to next code bit
  --
• decode Tree -gt Hcode -gt Char
• decode tr decodetree tr
• where
• decodetree (Node f t1 t2) (Lrest)
• decodetree t1 rest
• decodetree (Node f t1 t2) (Rrest)
• decodetree t2 rest
• decodetree (Leaf ch f) rest
• ch(decodetree tr rest)

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Example

• codetree Node 3 (Leaf a 0)
• (Node 3 (Leaf b 1) (Leaf t
  2))
• -- assume a is most frequent, denoted by
  smallest --
• -- number --
• message R,L,L,R,R,R,R,L,R,R
• decode codetree message
• gt
• decodetree Node 3 t1 (Node 3 ..)
• R L,L,R,R,R,R,L,R,R
• gt decodetree (Node 3 (Leaf b 1) (Leaf t 2))
• L L,R,R,R,R,L,R,R
• gt decodetree ( Leaf b 1) LR,R,R,R,L,R,R
• gt b decodetree Node 3 (Leaf a 0) (Node 3
  ..)
• L
  R,R,R,R,L,R,R
• gt b decodetree (Leaf a 0)) R,R,R,R,L,R,R
• gt b a decodetree Node 3 (Leaf a 0)
  (Node 3 ..)
• 
  R,R,R,R,L,R,R

10

• We still have to make
• the code tree
• the code table
• (Next lecture!)