I'm writing software that necessitates calculating hundreds of thousands of distances between points (in this case, in 67-space).
The distance between two points $p$ and $q$ using euclidean metrics is:
$$D_{p, q} = \sqrt{(p_1 - q_1)^2 + (p_2 - q_2)^2 + \dots + (p_n - q_n)^2}$$
However, using rectilinear metrics:
$$D_{p, q} = |p_1 - q_1| + |p_2 - q_2| + \dots + |p_n - q_n|$$
I found that using euclidean metrics I need $3n$ arithmetic operations whereas with rectilinear I only need $2n-1$: using rectilinear metrics I use $n+1$ less operations, or it's roughly 34% less operations which means speedier software.
Can rectilinear metrics be used in an equivalent fashion to euclidean for my purposes? I care about the closeness of points and their ordering; if I have three arbitrary points $p$, $q$, and $r$, if $p$ is closer to $q$ than $r$ using euclidean metrics, is it also true for rectilinear metrics irrespective of point values?
More declarative:
$$\begin{align} \text{if } D_{eucl.\text{ }p, q} &< D_{eucl.\text{ }p, r}\\ \text{then } D_{rect.\text{ }p, q} &< D_{rect.\text{ }p, r} \end{align}$$
For all integer values $p_1 \dots p_n$, $q_1 \dots q_n$, $r_1 \dots r_n$?
No. Let $p$ be the origin (both metrics are translation-invariant, so we can always shift so that this is the case). Take $q = (1.1,0,\ldots,0)$. Then $D_{eucl}(p,q) = D_{rect}(p,q) = 1.1 $. Take $r = \frac{\sqrt{2}}{2}(1,1,0,\ldots,0)$. Then $D_{eucl}(p,r) = 1 < D_{eucl}(p,q)$. But $D_{rect}(p,r) = \sqrt{2} > D_{rect}(p,q)$.