1. On Constructing the Huffman-Code-Based Reversible Variable-Length Codes
Authors: Chien-Wu Tsai and Ja-Ling Wu
Source: IEEE Transactions on Communications, Vol. 49, No. 9, September 2001, pp
Presenter: Iuon-Chang Lin
Date: 2001/9/27

2. Motivation
Due to the variable-length codes (VLCs), even one single bit mismatch will cause a serious problem resulting from the propagation of errors.
The main purpose of the reversible variable length codes (RVLCs) is to provide the capability of forward and backward decoding.

3. Symmetrical VS Asymmetrical RVLC
Symbol
Probability
Huffman Code
Sym. RVLC
Asym. RVLC A B C D E
0.33
0.30
0.18
0.10
0.09 00 01 11
100
101 00 11
010
101
0110 00 01 11
1010
10010
Symmetrical: require only one code table
Asymmetrical: offer better efficiently than symmetrical because a more flexible bit assignment is allowed, although two types of coding table are necessary

4. The Symmetrical RVLCs Constructions
0, 1
00, 11
000, 010, 101, 111
0000, 0110, 1001, 1111
00000, 01010, 00100, 01110, 10101, 11011, 10001, 11111
Level 1
Level 3
Level 4
Level 5
Level 6
Level 2
000000, , , , , , ,

5. Simple Symmetric CodingK L I J H G F E D C B A
Huffman Code
Simple Symm Code
Symbol
Prob. A B C D E F G H I J K L
0.32
0.18
0.08
0.07
0.06
0.05
0.04 11
101
1001
1000
0111
0110
0101
0100
0011
0010
0001
0000 00 11
010
101
0110
1001
01110
10001
011110
100001
0, 1
00, 11
000, 010, 101, 111
0000, 0110, 1001, 1111
00000, 01010, 00100, 01110, 10101, 11011, 10001, 11111
Level 1
Level 3
Level 4
Level 5
Level 6
Level 2
000000, , , , , , , × × × × × × × × × × × × × × × ×

6. Symmetric RVLC Algorithm

7. Symmetrical RVLC × × × × × × × × × × Huffman Code Symm. RVLC A B C D EK L I J H G F E D C B A
Huffman Code
Symbol
Prob.
Symm. RVLC A B C D E F G H I J K L
0.32
0.18
0.08
0.07
0.06
0.05
0.04 11
101
1001
1000
0111
0110
0101
0100
0011
0010
0001
0000 11
010
0000
0110
1001
10101
01110
10001
00100
011110
100001
001100
0, 1
00, 11
000, 010, 101, 111
0000, 0110, 1001, 1111
00000, 01010, 00100, 01110, 10101, 11011, 10001, 11111
Level 1
Level 3
Level 4
Level 5
Level 6
Level 2
000000, , , , , , , × × × × × × × × × ×

8. The Maximum Length of Symmetrical Bit Suffixes
0000000 level 3 level 4
01001010 level 3level 5
01110 level 5level 9
10101 level 5level 7

9. Asymmetrical RVLC Algorithm

10. Asymmetrical RVLC × × × × × × × × × × × × × × × × × × × × × × × × × ×
Huffman Code
0, 1
Symbol
Prob.
Asymm. RVLC
Level 1
Level 2
00, 01, 10, 11 A B C D E F G H I J K L
0.32
0.18
0.08
0.07
0.06
0.05
0.04 11
101
1001
1000
0111
0110
0101
0100
0011
0010
0001
0000 11
000
0010
0100
0101
0110
1001
1010
01110
10001
001100
011110 × × ×
Level 3
000, 001, 010, 011, 100, 101, 110, 111 × × ×
Level 4
0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 × × × × × × × × × × ×
00000, 00001, 00010, 00011, 00100, 00101, 00110, 00111, 01000, 01001, 01010, 01011, 01100, 01101, 01110, 01111, , 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111
Level 5 × × × × × × × × × × × × × × × × × × × × × × × × × ×

11. The Minimum Repetition Gap and No. of Available Candidate Codewords
Given a codeword c=010, B=010X1X2X3
X=101010101; B1=B3=B5=0; B2=B5=1; Gap=2.
X =010010010; B1=B4=0; B2=B5=1; B3=B6=2; Gap=3.
MRG=min{2,3}=2.

12. Symmetrical and Asymmetrical RVLCs for the English Alphabet

13. Encoding Symmetrical RVLC for English Alphabet
Step 1
Step 2
Step 3
Initialize
avail(3)=4
nrev(3)<avail(3)
avail(4)=4
nrev(4)>avail(4)
nrev(3)=n(3)=2
nrev(4)=n(4)=7
nrev(5)=n(5)=7
nrev(6)=n(6)=5
nrev(7)=n(7)=1
nrev(8)=n(8)=1
nrev(9)=n(9)=1
nrev(10)=n(10)=2
Select all
Select 010, 101
nrev(5)= nrev(5) + nrev(4)-avail(4)=10
nrev(4)=avail(4)=4
nrev(i): bit length vector of symm. RVLC; n(i): bit length vector of VLC
avail(i): the number of candidate codeword at level i

14. Encoding Symmetrical RVLC for English Alphabet (cont.)
Step 4
Step 5
Avail(5)=4
Avail(6)=4
nrev(5)>avail(5)
10>4
Select all
nrev(6)>avail(6)
11>4
Select all × × × × × × × ×
nrev(6)= nrev(6) + nrev(5)-avail(5)=11
nrev(5)=avail(5)=4
nrev(7)= nrev(7) + nrev(6)-avail(6)=8
nrev(6)=avail(6)=4

15. Encoding Asymmetrical RVLC
Step 1
Step 2
Step 3
Initialize
avail(3)=8
nrev(3)<avail(3)
2<8
avail(4)=10
nrev(4)<avail(4)
nrev(3)=n(3)=2
nrev(4)=n(4)=7
nrev(5)=n(5)=7
nrev(6)=n(6)=5
nrev(7)=n(7)=1
nrev(8)=n(8)=1
nrev(9)=n(9)=1
nrev(10)=n(10)=2
Select all
Select 000, 111
nrev(i): bit length vector of symm. RVLC; n(i): bit length vector of VLC
avail(i): the number of candidate codeword at level i

16. Encoding Asymmetrical RVLC (cont.)
Step 4
Step 5
Avail(5)=4
Avail(6)=3
nrev(5)>avail(5)
10>4
Select all
nrev(6)>avail(6)
11>3
Select all
nrev(6)= nrev(6) + nrev(5)-avail(5)=11
nrev(5)=avail(5)=4
nrev(7)= nrev(7) + nrev(6)-avail(6)=9
nrev(6)=avail(6)=3

17. Conclusion
The authors propose two proper and efficient codeword selection policies for symmetrical and asymmetrical classes.
It would let the number of available candidate as many as possible and coding overhead is as small as possible.
The polices are not the optimal one.