Truncated binary encoding is an entropy encoding typically used for uniform probability distributions with a finite alphabet. It is parameterized by an alphabet with total size of number n. It is a slightly more general form of binary encoding when n is not a power of two.
Contents
If n is a power of two then the coded value for 0 ≤ x < n is the simple binary code for x of length log2(n). Otherwise let k = floor( log2(n) ) such that 2k < n < 2k+1 and let u = 2k+1 - n.
Truncated binary encoding assigns the first u symbols codewords of length k and then assigns the remaining n - u symbols the last n - u codewords of length k + 1. Because all the codewords of length k + 1 consist of an unassigned codeword of length k with a "0" or "1" appended, the resulting code is a prefix code.
Example with n = 5
For example, for the alphabet {0, 1, 2, 3, 4}, n = 5 and 22 ≤ n < 23, hence k = 2 and u = 23 - 5 = 3. Truncated binary encoding assigns the first u symbols the codewords 00, 01, and 10, all of length 2, then assigns the last n - u symbols the codewords 110 and 111, the last two codewords of length 3.
For example, if n is 5, plain binary encoding and truncated binary encoding allocates the following codewords. Digits shown struck are not transmitted in truncated binary.
It takes 3 bits to encode n using straightforward binary encoding, hence 23 - n = 8 - 5 = 3 are unused.
In numerical terms, to send a value x where 0 ≤ x < n, and where there are 2k ≤ n < 2k+1 symbols, there are u = 2k + 1 − n unused entries when the alphabet size is rounded up to the nearest power of two. The process to encode the number x in truncated binary is: If x is less than u, encode it in k binary bits. If x is greater than or equal to u, encode the value x + u in k + 1 binary bits.
Example with n = 10
Another example, encoding an alphabet of size 10 (between 0 and 9) requires 4 bits, but there are 24 − 10 = 6 unused codes, so input values less than 6 have the first bit discarded, while input values greater than or equal to 6 are offset by 6 to the end of the binary space. (Unused patterns are not shown in this table.)
To decode, read the first k bits. If they encode a value less than u, decoding is complete. Otherwise, read an additional bit and subtract u from the result.
Example with n = 7
Here is a more extreme case: with n = 7 the next power of 2 is 8 so k = 2 and u = 23 - 7 = 1:
This last example demonstrates that a leading zero bit does not always indicate a short code; if u < 2k, some long codes will begin with a zero bit.
Simple algorithm
Generate the truncated binary encoding for a value x, 0 <= x < n, where n > 0 is the size of the alphabet containing x. n need not be a power of two.
string TruncatedBinary (int x, int n){ // Set k = floor(log2(n)), i.e., k such that 2^k <= n < 2^(k+1). int k = 0, t = n; while (t > 1) { k++; t >>= 1; } // Set u to the number of unused codewords = 2^(k+1) - n. int u = (1 << k+1) - n; if (x < u) return Binary(x, k); else return Binary(x+u, k+1));}The routine Binary is expository; usually just the rightmost len bits of the variable x are desired. Here we simply output the binary code for x using len bits, padding with high-order 0's if necessary.
string Binary (int x, int len){ string s = ""; while (x != 0) { if (even(x)) s = '0' + s; else s = '1' + s; x >>= 1; } while (s.Length < len) s = '0' + s; return s;}