// Levenshtein Coding - Recursive universal code with asymptotic optimality
Encodes length of length recursively until reaching 0.
Approaches theoretical minimum for large integers.
Works for any non-negative integer without parameters.
Levenshtein coding (also called Levenstein or L* coding) recursively encodes the bit length of a number. For integer n with binary length N: encode C(N-1) recursively, append '1', then append n without its leading 1. The recursion bottoms out at 0. This creates asymptotically optimal codes.
0 → 0 1 → 10 (C(0) + 1 + '') 2 → 110 (C(1) + 1 + '0') 3 → 111 (C(1) + 1 + '1') 4 → 11000 (C(2) + 1 + '00') 5 → 11001 (C(2) + 1 + '01') Recursive structure: C(0) = '0' C(1) = C(0) + 1 = '01' → '1' C(2) = C(1) + 1 + '0' = '110' C(3) = C(1) + 1 + '1' = '111'
Levenshtein coding (not to be confused with Levenshtein distance) is a universal code that recursively encodes the length of integers. Developed by Vladimir Levenshtein, it achieves asymptotic optimality for large integers.
The code recursively encodes the bit length minus one. To encode n with length N bits, we first encode (N-1), then append '1', then the last (N-1) bits of n. The recursion stops at 0, which encodes to '0'.
Levenshtein coding is asymptotically optimal like Elias Omega but uses a different recursive structure. It's more complex than Elias Gamma/Delta but achieves better compression for very large integers.
Levenshtein coding is primarily of theoretical interest in information theory and Kolmogorov complexity. It's rarely used in practice due to complexity, but demonstrates optimal universal coding principles.