Dynamic Huffman Dynamic Huffman CodingCoding

Computer Networks Computer Networks AssignmentAssignment

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TT Stage 1 (First occurrence of Stage 1 (First occurrence of t t ) ) rr / \/ \ 0 0 tt(1)(1)

Order: 0,Order: 0,tt(1)(1) ** r represents the root r represents the root** 0 represents the null node 0 represents the null node** t(1) denotes the occurrence of t(1) denotes the occurrence of TT with a frequency with a frequency

of 1 of 1

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TETE

Stage 2 (First occurrence of Stage 2 (First occurrence of ee)) rr / \/ \ 1 1 tt(1)(1) / \/ \ 0 0 ee(1)(1)

Order: 0,Order: 0,ee(1),1,(1),1,tt(1)(1)

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TENTEN Stage 3 (First occurrence of Stage 3 (First occurrence of nn ) ) rr / \/ \ 2 2 tt(1)(1) / \/ \ 1 1 ee(1)(1) / \ / \ 0 0 nn(1)(1)

Order: 0,Order: 0,nn(1),1,(1),1,ee(1),2,(1),2,tt(1) : (1) : MisfitMisfit

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Reorder: TENReorder: TEN

rr / \/ \ tt(1) 2 (1) 2 / \/ \ 1 1 ee(1)(1) / \/ \ 0 0 nn(1)(1)

Order: 0,Order: 0,nn(1),1,(1),1,ee(1),(1),tt(1),2(1),2

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TENNTENN Stage 4 ( Repetition of Stage 4 ( Repetition of nn ) ) rr / \/ \ tt(1) 3(1) 3 / \/ \ 2 2 ee(1)(1) / \/ \ 0 0 nn(2)(2)

Order: 0,Order: 0,nn(2),2,(2),2,ee(1),(1),tt(1),3 : (1),3 : MisfitMisfit

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Reorder: TENNReorder: TENN

rr / \/ \ nn(2) 2(2) 2 / \/ \ 1 1 ee(1)(1) / \/ \ 0 0 tt(1)(1) Order: 0,Order: 0,tt(1),1,(1),1,ee(1),(1),nn(2),2 (2),2 tt(1),(1),nn(2) are swapped(2) are swapped

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TENNETENNE Stage 5 (Repetition of Stage 5 (Repetition of ee ) ) rr / \/ \ nn(2) 3(2) 3 / \/ \ 1 1 ee(2)(2) / \/ \ 0 0 tt(1)(1)

Order: 0,Order: 0,tt(1),1,(1),1,ee(2),(2),nn(2),3 (2),3

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TENNESTENNES Stage 6 (First occurrence of s)Stage 6 (First occurrence of s) rr / \/ \ nn(2) 4(2) 4 / \/ \ 2 2 ee(2)(2) / \/ \ 1 1 tt(1)(1) / \/ \ 0 0 ss(1)(1)

Order: 0,Order: 0,ss(1),1,(1),1,tt(1),2,(1),2,ee(2),(2),nn(2),4(2),4

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TENNESSTENNESS Stage 7 (Repetition of Stage 7 (Repetition of ss)) rr / \/ \ nn(2) 5(2) 5 / \/ \ 3 3 ee(2)(2) / \/ \ 2 2 tt(1)(1) / \/ \ 0 0 ss(2)(2)

Order: 0,Order: 0,ss(2),2,(2),2,tt(1),3,(1),3,ee(2),(2),nn(2),5 : (2),5 : MisfitMisfit

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Reorder: TENNESSReorder: TENNESS

rr / \/ \ n(2) 5n(2) 5 / \/ \ 3 e(2)3 e(2) / \/ \ 1 s (2)1 s (2) / \/ \ 0 t(1) 0 t(1)

Order : 0,t(1),1,s(2),3,e(2),n(2),5Order : 0,t(1),1,s(2),3,e(2),n(2),5 s(2) and t(1) are swapped s(2) and t(1) are swapped

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TENNESSETENNESSE

Stage 8 (Second repetition of Stage 8 (Second repetition of ee ) ) rr / \/ \ nn(2) 6(2) 6 / \/ \ 3 3 ee(3)(3) / \/ \ 1 1 ss(2)(2) / \/ \ 0 0 tt(1)(1)

Order : 0,Order : 0,tt(1),1,(1),1,ss(2),3,(2),3,ee(3),(3),nn(2),6 : (2),6 : MisfitMisfit

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Reorder: TENNESSEReorder: TENNESSE

rr / \/ \ ee(3) 5(3) 5 / \/ \ 3 3 nn(2)(2) / \/ \ 1 1 ss(2)(2) / \/ \ 0 0 tt(1)(1) Order : 1,t(1),1,s(2),3,n(2),e(3),5Order : 1,t(1),1,s(2),3,n(2),e(3),5 N(2) and e(3) are swappedN(2) and e(3) are swapped

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TENNESSEETENNESSEE

Stage 9 (Second repetition of Stage 9 (Second repetition of ee ) ) rr 00/ \/ \11

ee(4) 5(4) 5 00/ \/ \11

3 3 nn(2)(2) 00/ \/ \11

1 1 ss(2)(2) 00/ \/ \11

0 0 tt(1)(1) Order : 1,Order : 1,tt(1),1,(1),1,ss(2),3,(2),3,nn(2),(2),ee(4),5(4),5

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ENCODINGENCODING

The letters can be encoded as follows:The letters can be encoded as follows:

e : 0e : 0 n : 11n : 11 s : 101s : 101 t : 1001t : 1001

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Average Code LengthAverage Code Length

Average code length = Average code length = i=0,n i=0,n (length*frequency)/ (length*frequency)/ i=0,ni=0,n

frequencyfrequency

= { 1(4) + 2(2) + 3(2) + 1(4) } / = { 1(4) + 2(2) + 3(2) + 1(4) } / (4+2+2+1)(4+2+2+1)

= 18 / 9 = 2= 18 / 9 = 2

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ENTROPYENTROPY

Entropy = -Entropy = -i=1i=1,,nn (p (pii loglog2 2 ppii))

= - ( 0.44 * log= - ( 0.44 * log220.44 + 0.22 * log0.44 + 0.22 * log220.220.22

+ 0.22 * log+ 0.22 * log220.22 + 0.11 * 0.22 + 0.11 * loglog220.11 )0.11 )

= - (0.44 * log0.44 + 2(0.22 * log0.22 + 0.11 * = - (0.44 * log0.44 + 2(0.22 * log0.22 + 0.11 * log0.11)log0.11)

/ log2/ log2

= 1.8367= 1.8367

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Ordinary Huffman CodingOrdinary Huffman Coding

TENNESSETENNESSE 99 00/ \/ \11

5 5 ee(4)(4) 00/ \/ \11

ss(2) 3(2) 3 00/ \/ \11

tt(1) (1) nn(2) (2)

ENCODINGENCODING E : 1E : 1 S : 00S : 00 T : 010T : 010 N : 011N : 011

Average code length = (1*4 Average code length = (1*4 + 2*2 + 2*3 + 3*1) / 9 = + 2*2 + 2*3 + 3*1) / 9 = 1.891.89

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SUMMARYSUMMARY

The average code length of ordinary Huffman The average code length of ordinary Huffman coding seems to be better than the Dynamic version,in this coding seems to be better than the Dynamic version,in this exercise.exercise.

But, actually the performance of dynamic coding is But, actually the performance of dynamic coding is better. The problem with static coding is that the tree has better. The problem with static coding is that the tree has to be constructed in the transmitter and sent to the to be constructed in the transmitter and sent to the receiver. The tree may change because the frequency receiver. The tree may change because the frequency distribution of the English letters may change in plain text distribution of the English letters may change in plain text technical paper, piece of code etc. technical paper, piece of code etc.

Since the tree in dynamic coding is constructed on the Since the tree in dynamic coding is constructed on the receiver as well, it need not be sent. Considering this, receiver as well, it need not be sent. Considering this, Dynamic coding is better.Dynamic coding is better.

Also, the average code length will improve if the Also, the average code length will improve if the transmitted text is bigger.transmitted text is bigger.