Sunday 3 May 2020

Information on incrementation

Defining increment

Just to avoid any confusion, the operation that this post is about is adding 1 (one) to a value: $$\text{increment}(x) = x + 1$$ Specifically, performing that operation in the domain of bit-vectors.

Incrementing is very closely related to negating. After all, -x = ~x + 1 and therefore x + 1 = -~x, though putting it that way feel oddly reversed to me.

Bit-string notation

In bit-string notation (useful for analysing compositions of operations at the bit level), increment can be represented as: $$a01^k + 1 = a10^k$$

An "English" interpretation of that form is that an increment carries through the trailing set bits, turning them to zero, and then carries into the right-most unset bit, setting it.

That "do something special with the right-most unset bit" aspect of increment is the basis for various right-most bit manipulations, some of which were implemented in AMD Trailing Bit Manipulation (TBM) (which has been discontinued).

For example, the right-most unset bit in x can be set using x | (x + 1), which has a nice symmetry with the more widely known trick for unsetting the right-most set bit, x & (x - 1).

Increment by XOR

As was the case with negation, there is a way to define increment in terms of XOR. The bits that flip during an increment are all the trailing set bits and the right-most unset bit, the TBM instruction for which is BLCMSK. While that probably does not seem very useful yet, the fact that x ^ (x + 1) takes the form of some number of leading zeroes followed by some number of trailing ones turns out to be useful.

Suppose one wants to increment a bit-reversed integer, a possible (and commonly seen) approach is looping of the bits from top the bottom and implementing the "carry through the ones, into the first zero" logic by hand. However, if the non-reversed value was also available (let's call it i), the bit-reversed increment could be implemented by calculating the number of ones in the mask as tzcnt(i + 1) + 1 (or popcnt(i ^ (i + 1))) and forming a mask with that number of ones located at the desired place within an integer:

// i   = normal counter
// rev = bit-reversed counter
// N   = 1 << number_of_bits
int maskLen = tzcnt(i + 1) + 1;
rev ^= N - (N >> maskLen);
That may still not seem useful, but this enables an implementation of the bit-reversal permutation (not a bit-reversal itself, but the permutation that results from bit-reversing the indices). The bit-reversal permutation is sometimes used to re-order the result of a non-auto-sorting Fast Fourier Transform algorithm into the "natural" order. For example,
// X = array of data
// N = length of X, power of two
for (uint32_t i = 0, rev = 0; i < N; ++i)
    if (i < rev)
        swap(X[i], X[rev]);
    int maskLen = tzcnt(i + 1) + 1;
    rev ^= N - (N >> maskLen);
This makes no special effort to be cache-efficient.