Huffman coding

Huffman coding

Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 2

Optimal codes - I

A code is optimal if it has the shortest codeword length L

This can be seen as an optimization problem1

m

i ii

L p l=

=∑

1

1

min

subject to 1i

m

i ii

ml

i

l p

D

=

−

=

≤

∑

∑


Optimal codes - II

Let’s make two simplifying assumptionsno integer constraint on the codelengthsKraft inequality holds with equality

Lagrange-multiplier problem

1 1

1im m

li i

i i

J p l Dλ −= =

⎛ ⎞= + −⎜ ⎟⎝ ⎠

∑ ∑

0 log 0 log

j jl l jj

j

pJ p D D Dl D

λλ

− −∂ = → − = → =∂


Optimal codes - III

Substitute into the Kraft inequality

that is

Note that

logjl jpD

Dλ− =

1

11 log log

i

mli

ii

p p DD D

λλ

−

=

= → = → =∑* logi D il p= −

** log ( ) !!m m

i i i D i Dp l p pL H X= == −∑ ∑

the entropy, when we use base D for logarithms

1 1i i= =


Optimal codes - IV

In practice the codeword lengths must be integer value, so obtained results is a lower bound

TheoremThe expected length of any istantaneous D-ary code for a r.v. X satisfies

this fundamental result derives frow the work of Shannon

( )DL H x≥


Optimal codes - V

What about the upper bound?

TheoremGiven a source alphabet (i.e. a r.v.) of entropy it is possible to find an instantaneous binary code which length satisfies

A similar theorem could be stated if we use the wrong probabilities instead of the true ones ; the only difference is a term which accounts for the relative entropy

( )H X

( ) ( ) 1H X L H X≤ ≤ +

{ }ip{ }iq


The redundance

It is defined as the average codeword legths minus the entropy

Note that

(why?)

Redundancy logi ii

L p p⎛ ⎞= − −⎜ ⎟⎝ ⎠∑

0 redundancy 1≤ ≤


Compression ratio

It is the ratio between the average number of bit/symbol in the original message and the same quantity for the coded message, i.e.

average original symbol lengthaverage compressed symbol length

C < >=< >

( )!!L X≠


Uniquely decodable codes

The set of the instantaneous codes are a small subset of the uniquely decodable codes. It is possible to obtain a lower average code length L using a uniquely decodable code that is not

instantaneous? NOSo we use instantaneous codes that are easier to decode


Summary

Average codeword length Lfor uniquely decodable codes

(and for instantaneous codes)In practice for each r.v. with entropy we can build a code with average codeword length that satisfies

( )L H X≥

( )H XX

( ) ( ) 1H X L H X≤ ≤ +


Shannon-Fano codingThe main advantage of the Shannon-Fano technique is its semplicity

Source symbols are listed in order of nonincreasing probability.The list is divided in such a way to form two groups of as nearly equal probabilities as possibleEach symbol in the first group receives a 0 as first digit of its codeword, while the others receive a 1Each of these group is then divided according to the same criterion and additional code digits are appendedThe process is continued until each group contains only one message


example

1 2 1 4 1 8 1 16 1 32 1 32

abcdef

011111

01111

0111

011

01

H=1.9375 bits

L=1.9375 bits


Shannon-Fano coding - exercise

Symb. Prob. * 12% ? 5% ! 13% & 2% $ 29% € 13% § 10% ° 6% @ 10%

Encode, using Shannon-Fano algorithm


Is Shannon-Fano coding optimal?

0.35 0.17 0.17 0.16 0.15

abcde

0100101110111

000110110111

H=2.2328 bits

L=2.31 bits

L1=2.3 bits


Huffman coding - I

There is another algorithm which performances are slightly better than Shanno-Fano, the famous Huffman codingIt works constructing bottom-up a tree, that has symbols in the leafsThe two leafs with the smallest probabilities becomes sibling under a parent node with probabilities equal to the two children’s probabilities


Huffman coding - II

At this time the operation is repeated, considering also the new parent node and ignoring its childrenThe process continue until there is only parent node with probability 1, that is the root of the treeThen the two branches for every non-leaf node are labeled 0 and 1 (typically, 0 on the left branch, but the order is not important)


Huffman coding - example

0

Symbol Prob. 0.05 0.05 0.1 0.2 0.3 0.2 0.1

abcdefg a

0.05b

0.05c

0.1d

0.2e

0.3f

0.2g

0.1

0.1

0.2

0.3

0.4

0.6

1.00

0

0

0

0

1

1

1

1

1

1

a0.05

b0.05

c0.1

d0.2

e0.3

f0.2

g0.1

0.1

0.2

0.3

0.4

0.6

1.0


Huffman coding - example

Symbol Prob. Codeword 0.05 0000 0.05 0001 0.1 001 0.2 01 0.3 10 0.2 11

abcdef 0

0.1 111g

Exercise: evaluate H(X) and L(X)

H(X)=2.5464 bits

L(X)=2.6 bits !!


Huffman coding - exercise

Code the sequence

aeebcddegfced and calculate the compression ratio

Sol: 0000 10 10 0001 001 01 01

10 111 110 001 10 01

Aver. orig. symb. length = 3 bits

Aver. compr. symb. length = 34/13

C=.....


abcdef 0

0.1 111g



Decode the sequence0111001001000001111110

Sol: dfdcadgf


abcdef 0

0.1 111g



Encode with Huffman the sequence01$cc0a02ba10

and evaluate entropy, average codeword length and compression ratio

Symb. Prob. 0.10 0.03 0.14 0 0.4 1 0.22 2 0.04 $ 0.07

abc



Symb. Prob. 0 0.16 1 0.02 2 0.15 3 0.29 4 0.17 5 0.04 % 0.17

Decode (if possible) the Huffman coded bit streaming01001011010011110101...


Huffman coding - notes

In the huffman coding, if, at any time, there is more than one way to choose a smallest pair of probabilities, any such pair may be chosen

Sometimes, the list of probabilities is inizialized to be non-increasing and reordered after each node creation. This details doesn’t affect the correctness of the algorithm, but it provides a more efficient implementation



There are cases in which the Huffman coding does not uniquely determine codeword lengths, due to the arbitrary choice among equal minimum probabilities.For example for a source with probabilities

it is possible to obtain codeword lengths of and ofIt would be better to have a code which codelength has the minimum variance, as this solution will need the minimum buffer space in the transmitter and in the receiver

{ }0.4, 0.2, 0.2, 0.1, 0.1{ }1, 2, 3, 4, 4 { }2, 2, 2, 3, 3



Schwarz defines a variant of the Huffman algorithm that allows to build the code with minimum .

There are several other variants, we will explain the most important in a while.

maxl


Optimality of Huffman coding - I

It is possible to prove that, in case of character coding (one symbol, one codeword), Huffman coding is optimal

In another terms Huffman code has minimum redundancyAn upper bound for redundancy has been found

where is the probability of the most likely simbol

( )1 2 2 2 1redundancy 1 log log log 0.086p e e p≤ + − + +

1p


Optimality of Huffman coding - II

Why Huffman code “suffers” when there is one symbol with very high probability?Remember the notion of uncertainty...

The main problem is given by the integer constraint on codelengths!!

This consideration opens the way to a more powerful coding... we will see it later

( ) 1 log( ( )) 0p x p x→ ⇒ − →


Huffman coding - implementation

Huffman coding can be generated in O(n) time, where n is the number of source symbols, provided that probabilities have been presorted (however this sort costs O(nlogn)...)

Nevertheless, encoding is very fast



However, spatial and temporal complexity of the decoding phase are far more important, because, on average, decoding will happen more frequently.Consider a Huffman tree with n symbols

n leafs and n-1 internal nodes

has the pointer to a symbol and the info that it is a leaf

has two pointers

2 2( 1) 4 words (32 bits)n n n+ −



1 million symbols 16 MB of memory!Moreover traversing a tree from root to leaf involves follow a lot of pointers, with little locality of reference. This causes several page faults or cache misses.

To solve this problem a variant of Huffman coding has been proposed: canonical Huffman coding


canonical Huffman coding - I

Symb. Prob. Code 1 Code 2 Code 3 0.11 000 0.12 001 0.13 100

1111

0000010

1001

10

01

abcd .14 101

0.24 01 0.26 11

0101000

011

10

1 1ef

b0.12

c0.13

d0.14

e0.24

f0.26

a0.11

0.23 0.27

0.470.53

1.0

00

1

1 1

1

(0)

(0)

(0)

(0)(0)

(1)

(1)(1)

(1) (1)

?

10

00


canonical Huffman coding - IIThis code cannot be obtained through a Huffman tree!

We do call it an Huffman code because it is instantaneous and the codeword lengths are the same than a valid Huffman code

numerical sequence propertycodewords with the same length are ordered lexicographicallywhen the codewords are sorted in lexical order they are also in order from the longest to the shortest codeword

Symb. Code 3

000001010011

10 1 1

abcdef


canonical Huffman coding - III

The main advantage is that it is not necessary to store a tree, in order to decodingWe need

a list of the symbols ordered according to the lexical order of the codewordsan array with the first codeword of each distinct length

34

canonical Huffman coding - IVEncoding. Suppose there are n disctinct symbols, that for symbol

i we have calculated huffman codelength andil ii l maxlength∀ ≤for 1 to { [ ] 0; }for 1 to { [ ] [ ] 1; }

[ ] 0;for 1 downto 1 { [ ] ( [ 1] [ 1]) / 2 ; }

for 1 to

i i

k maxlength numl ki n numl l numl l

firstcode maxlengthk maxlength

firstcode k firstcode k numl kk maxlength

= == = +

== −

= + + +⎡ ⎤⎢ ⎥=

[ ]

{ [ ]= [ ]; }for 1 to { [ ] [ ]; , [ ] - [ ] ; [ ] [ ] 1; }

i

i i i

i i

nextcode k firstcode ki n

codeword i nextcode lsymbol l nextcode l firstcode l inextcode l nextcode l

==

=

= +

numl[k] = number of codewords with length k

firstcode[k] = integer for first code of length k

nextcode[k] = integer for the next codeword of length k to be assigned

symbol[-,-] used for decoding

codeword[i] the rightmost bits of this integer are the code for symbol i

il

35

canonical Huffman - example

1. Evaluate array numlSymb. length 2 5 5 3 2 5 5 2

ii labcdefgh

: [0 3 1 0 4]numl

2. Evaluate array firstcode

: [2 1 1 2 0]firstcode3. Construct array codeword and symbol

[ ]

for 1 to { [ ]= [ ]; }for 1 to { [ ] [ ]; , [ ]- [ ] ; [ ] [ ] 1; }

i

i i i

i i

k maxlengthnextcode k firstcode ki n

codeword i nextcode lsymbol l nextcode l firstcode l inextcode l nextcode l

=

==

=

= +

- - - -

a e h -

d - - -

- - - -

b c f g

symbol0 1 2 3

1

2

3

4

5

code bitsword 1 01 0 00000 1 00001 1 001 2 10 2 00010 3 00011 3 11

for 1 downto 1 {

[ ] ( [ 1]

[ 1]) / 2 ; }

k maxlength

firstcode k firstcode k

numl k

= −

= + +

+ +


canonical Huffman coding - VDecoding. We have the arrays firstcode and symbols

[ ]

();1;

while [ ] { 2* (); 1; }Return , [ ] ;

v nextinputbitk

v firstcode kv v nextinputbitk k

symbol k v firstcode k

==

<= += +

−

nextinputbit() function that returns next input bit

firstcode[k] = integer for first code of length k

symbol[k,n] returns the symbol number n with codelength k

37

canonical Huffman - example

[ ]

();1;

while [ ] { 2* (); 1; }Return , [ ] ;

v nextinputbitk

v firstcode kv v nextinputbitk k

symbol k v firstcode k

==

<= += +

−

- - - -

a e h -

d - - -

- - - -

b c f g

symbol0 1 2 3

1

2

3

4

5: [2 1 1 2 0]firstcode

00 00 00 000 0011 11 1100 00 00 000 0011 11 11

symbol[3,0] = dsymbol[2,2] = hsymbol[2,1] = esymbol[5,0] = bsymbol[2,0] = asymbol[3,0] = d

symbol[3,0] = dsymbol[2,2] = hsymbol[2,1] = esymbol[5,0] = bsymbol[2,0] = asymbol[3,0] = d

Decoded: dhebad

Huffman codingOptimal codes - IOptimal codes - IIOptimal codes - IIIOptimal codes - IVOptimal codes - VThe redundanceCompression ratioUniquely decodable codesSummaryShannon-Fano codingexampleShannon-Fano coding - exerciseIs Shannon-Fano coding optimal?Huffman coding - IHuffman coding - IIHuffman coding - exampleHuffman coding - exampleHuffman coding - exerciseHuffman coding - exerciseHuffman coding - exerciseHuffman coding - exerciseHuffman coding - notesHuffman coding - notesHuffman coding - notesOptimality of Huffman coding - IOptimality of Huffman coding - IIHuffman coding - implementationHuffman coding - implementationHuffman coding - implementationcanonical Huffman coding - Icanonical Huffman coding - IIcanonical Huffman coding - IIIcanonical Huffman coding - IVcanonical Huffman - examplecanonical Huffman coding - Vcanonical Huffman - example

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