Calculate distance between two vectors of different length
There are different methods to calculate distance between two vectors of the same length: Euclidean, Manhattan, Hamming ...
I'm wondering about any method that would calculate distance between vectors of different length.
Answer
The Euclidean distance formula finds the distance between any two points in Euclidean space.
A point in Euclidean space is also called a Euclidean vector.
You can use the Euclidean distance formula to calculate the distance between vectors of two different lengths.
For vectors of different dimension, the same principle applies.
Suppose a vector of lower dimension also exists in the higher dimensional space. You can then set all of the missing components in the lower dimensional vector to 0 so that both vectors have the same dimension. You would then use any of the mentioned distance formulas for computing the distance.
For example, consider a 2-dimensional vector A in R² with components (a1,a2), and a 3-dimensional vector B in R³ with components (b1,b2,b3).
To express A in R³, you would set its components to (a1,a2,0). Then, the Euclidean distance d between A and B can be found using the formula:
d² = (b1 - a1)² + (b2 - a2)² + (b3 - 0)²
d = sqrt((b1 - a1)² + (b2 - a2)² + b3²)
For your particular case, the components will be either 0 or 1, so all differences will be -1, 0, or 1. The squared differences will then only be 0 or 1.
If you're using integers or individual bits to represent the components, you can use simple bitwise operations instead of some arithmetic (^ means XOR or exclusive or):
d = sqrt(b1 ^ a1 + b2 ^ a2 + ... + b(n-1) ^ a(n-1) + b(n) ^ a(n))
And we're assuming the trailing components of A are 0, so the final formula will be:
d = sqrt(b1 ^ a1 + b2 ^ a2 + ... + b(n-1) + b(n))