Minkowski distance

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},\ldots ,x_{n}){\text{ and }}Y=(y_{1},y_{2},\ldots ,y_{n})\in {\mathbb {R}}^{n}

is defined as:

\left(\sum_{i=1}^n |x_i-y_i|^p\right)^{1/p}

For $p\geq 1$ , 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\to\infty}{\left(\sum_{i=1}^n |x_i-y_i|^p\right)^\frac{1}{p}} = \max_{i=1}^n |x_i-y_i|. \,

Similarly, for p reaching negative infinity, we have:

\lim_{p\to-\infty}{\left(\sum_{i=1}^n |x_i-y_i|^p\right)^\frac{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:

Unit circles using different Minkowski distance metrics.

External links

Simple IEEE 754 implementation in C++

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Minkowski distance

Definition

See also

External links