Minkowski distance

Reading time: 15 minutes

Minkowski distance is a distance/ similarity measurement between two points in the normed vector space (N dimensional real space) and is a generalization of the Euclidean distance and the Manhattan distance.

Minkowski distance in N-D space

In a N dimensional space, a point is represented as (x1, x2, ..., xN).

Consider two points P1 and P2:


P1: (X1, X2, ..., XN)
P2: (Y1, Y2, ..., YN)

Then, the Minkowski distance between P1 and P2 is given as:

$$ \sqrt[p]{{(x1-y1)}^p\ +\ {(x2-y2)}^p\ +\ ...\ +\ {(xN-yN)}^p}
$$

Euclidean distance from Minkowski distance

When p = 2, Minkowshi distance is same as Euclidean distance.

Manhattan distance from Minkowski distance

When p = 1, Minkowshi distance is same as Manhattan distance.

Visualize

Unit circles (path represents points with same Minkowshi distance) with various values of p (Minkowski distance):

opengenus_minkowshi_distance

Applications

Applications of Minkowshi Distance are:

Euclidean distance and Manhattan distance are same as Minkowshi Distance, hence, the applications of the previous two distance metrics are applications of Minkowshi Distance.