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Levenshtein coding

Levenshtein coding is a universal code encoding the non-negative integers developed by Vladimir Levenshtein.[1][2]

Encoding

The code of zero is "0"; to code a positive number:

  1. Initialize the step count variable C to 1.
  2. Write the binary representation of the number without the leading "1" to the beginning of the code.
  3. Let M be the number of bits written in step 2.
  4. If M is not 0, increment C, repeat from step 2 with M as the new number.
  5. Write C "1" bits and a "0" to the beginning of the code.

The code begins:

NumberEncodingImplied probability
001/2
1101/4
2110 01/16
3110 11/16
41110 0 001/128
51110 0 011/128
61110 0 101/128
71110 0 111/128
81110 1 0001/256
91110 1 0011/256
101110 1 0101/256
111110 1 0111/256
121110 1 1001/256
131110 1 1011/256
141110 1 1101/256
151110 1 1111/256
1611110 0 00 00001/4096
1711110 0 00 00011/4096

To decode a Levenshtein-coded integer:

  1. Count the number of "1" bits until a "0" is encountered.
  2. If the count is zero, the value is zero, otherwise
  3. Discard the "1" bits just counted and the first "0" encountered
  4. Start with a variable N, set it to a value of 1 and repeat count minus 1 times:
  5. 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

Encoding

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

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

References

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