Minkowski distance - Alchetron, The Free Social Encyclopedia

The Minkowski distance is a metric in a normed vector space which can be considered as a generalization of both the Euclidean distance and the Manhattan distance.

Definition

The Minkowski distance of order p between two points

X = ( x 1 , x 2 , … , x n ) and Y = ( y 1 , y 2 , … , y n ) ∈ R n

is defined as:

( ∑ i = 1 n | x i − y i | p ) 1 / p

For p ≥ 1 , the Minkowski distance is a metric as a result of the Minkowski inequality. When p < 1 , the distance between (0,0) and (1,1) is 2 1 / p > 2 , but the point (0,1) is at a distance 1 from both of these points. Since this violates the triangle inequality, for p < 1 it is not a metric.

Minkowski distance is typically used with p being 1 or 2. The latter is the Euclidean distance, while the former is sometimes known as the Manhattan distance. In the limiting case of p reaching infinity, we obtain the Chebyshev distance:

lim p → ∞ ( ∑ i = 1 n | x i − y i | p ) 1 p = max i = 1 n | x i − y i | .

Similarly, for p reaching negative infinity, we have:

lim p → − ∞ ( ∑ i = 1 n | x i − y i | p ) 1 p = min i = 1 n | x i − y i | .

The Minkowski distance can also be viewed as a multiple of the power mean of the component-wise differences between P and Q.

The following figure shows unit circles with various values of p:

References

Minkowski distance Wikipedia

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