Fastest way to scan for bit pattern in a stream of bits

Question

I need to scan for a 16 bit word in a bit stream. It is not guaranteed to be aligned on byte or word boundaries.

What is the fastest way of achieving this

Answer 1

You can use the fast fourier transform for extremely large input (value of n) to find any bit pattern in O(n log n ) time. Compute the cross-correlation of a bit mask with the input. Cross -correlation of a sequence x and a mask y with a size n and n' respectively is defined by

R(m) = sum  _ k = 0 ^ n' x_{k+m} y_k

then occurences of your bit pattern that match the mask exactly where R(m) = Y where Y is the sum of one's in your bit mask.

So if you are trying to match for the bit pattern

[0 0 1 0 1 0]

in

[ 1 1 0 0 1 0 1 0 0 0 1 0 1 0 1]

then you must use the mask

[-1 -1  1 -1  1 -1]

the -1's in the mask guarantee that those places must be 0.

You can implement cross-correlation, using the FFT in O(n log n ) time.

I think KMP has O(n + k) runtime, so it beats this out.