Algorithms
January 15, 2020

## Distance Measures ### Manhattan Distance

Manhattan distance calculates the distance between two real-valued vectors.

$$D_{Manhattan}(x,y) = \sum_{i=1}^{n} |x_i - y_i|$$

### Euclidean Distance

Euclidean distance calculates the distance between two real-valued vectors.

$$D_{Euclidean}(x,y) = \sqrt{\sum_{i=1}^{n} |x_i - y_i|^2}$$

### Minkowski Distance

Minkowski distance calculates the distance between two real-valued vectors.

It is a generalization of the Euclidean and Manhattan distance measures and adds a parameter, called the “order” or “p“, that allows different distance measures to be calculated.

• $$p$$ is an integer
• $$x=(x_1,x_2,…,x_n) ]\in \mathbb{R}^n$$ is a real-valued vector
• $$y=(y_1,y_2,…,y_n) ]\in \mathbb{R}^n$$ is a real-valued vector

$$D_{Minkowski}(x,y) = \Big( \sum_{i=1}^{n} | x_i - y_i | ^p \Big) ^{1/p}$$

• $$p=1 \Rightarrow$$ Manhattan distance
• $$p=2 \Rightarrow$$ Euclidean distance
• $$p=\infty \Rightarrow$$ Chebyshev distance