I am trying to simplify the euclidean distance function to the reduce computation time of some code. I am not interested on the numerical result of the distance but rather on which is the closest vector, but I do not care by how much (I hope that makes sense).
The best approximation I have found so far is the Manhattan distance. However, I would like to know if there is any established method of quantifying the error incurred by using the Manhattan distance.
What I had in mind is generating random vectors of the same dimension I use and quantifying the percentage of times the result of the comparison is different when using euclidean distance than Manhattan, as I am only interested in the result of the comparison but not on the value of the distance itself.