Levenshtein coding
From Wikipedia the free encyclopedia
Levenshtein coding is a universal code encoding the non-negative integers developed by Vladimir Levenshtein.[1][2]
Encoding
[edit]The code of zero is "0"; to code a positive number:
- Initialize the step count variable C to 1.
- Write the binary representation of the number without the leading "1" to the beginning of the code.
- Let M be the number of bits written in step 2.
- If M is not 0, increment C, repeat from step 2 with M as the new number.
- Write C "1" bits and a "0" to the beginning of the code.
The code begins:
| Number | Encoding | Implied probability | |
|---|---|---|---|
| 0 | 0 | 1/2 | |
| 1 | 10 | 1/4 | |
| 2 | 110 0 | 1/16 | |
| 3 | 110 1 | 1/16 | |
| 4 | 1110 0 00 | 1/128 | |
| 5 | 1110 0 01 | 1/128 | |
| 6 | 1110 0 10 | 1/128 | |
| 7 | 1110 0 11 | 1/128 | |
| 8 | 1110 1 000 | 1/256 | |
| 9 | 1110 1 001 | 1/256 | |
| 10 | 1110 1 010 | 1/256 | |
| 11 | 1110 1 011 | 1/256 | |
| 12 | 1110 1 100 | 1/256 | |
| 13 | 1110 1 101 | 1/256 | |
| 14 | 1110 1 110 | 1/256 | |
| 15 | 1110 1 111 | 1/256 | |
| 16 | 11110 0 00 0000 | 1/4096 | |
| 17 | 11110 0 00 0001 | 1/4096 | |
To decode a Levenshtein-coded integer:
- Count the number of "1" bits until a "0" is encountered.
- If the count is zero, the value is zero, otherwise
- Discard the "1" bits just counted and the first "0" encountered
- Start with a variable N, set it to a value of 1 and repeat count minus 1 times:
- Read N bits (and remove them from the encoded integer), prepend "1", assign the resulting value to N
The Levenshtein code of a positive integer is always one bit longer than the Elias omega code of that integer. However, there is a Levenshtein code for zero, whereas Elias omega coding would require the numbers to be shifted so that a zero is represented by the code for one instead.
Example code
[edit]Encoding
[edit]void levenshteinEncode(char* source, char* dest) { IntReader intreader(source); BitWriter bitwriter(dest); while (intreader.hasLeft()) { int num = intreader.getInt(); if (num == 0) bitwriter.outputBit(0); else { int c = 0; BitStack bits; do { int m = 0; for (int temp = num; temp > 1; temp>>=1) // calculate floor(log2(num)) ++m; for (int i=0; i < m; ++i) bits.pushBit((num >> i) & 1); num = m; ++c; } while (num > 0); for (int i=0; i < c; ++i) bitwriter.outputBit(1); bitwriter.outputBit(0); while (bits.length() > 0) bitwriter.outputBit(bits.popBit()); } } } Decoding
[edit]void levenshteinDecode(char* source, char* dest) { BitReader bitreader(source); IntWriter intwriter(dest); while (bitreader.hasLeft()) { int n = 0; while (bitreader.inputBit()) // potentially dangerous with malformed files. ++n; int num; if (n == 0) num = 0; else { num = 1; for (int i = 0; i < n-1; ++i) { int val = 1; for (int j = 0; j < num; ++j) val = (val << 1) | bitreader.inputBit(); num = val; } } intwriter.putInt(num); // write out the value } bitreader.close(); intwriter.close(); } See also
[edit]References
[edit]- ^ "1968 paper by V. I. Levenshtein (in Russian)" (PDF).
- ^ David Salomon (2007). Variable-length codes for data compression. Springer. p. 80. ISBN 978-1-84628-958-3.