Manhattan distance

The Manhattan distance is a different way of measuring distance. It is named after the grid shape of streets in Manhattan. If there are two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$, the Manhattan distance between the two points is $|x_{1}-x_{2}|+|y_{1}-y_{2}|$.

This distance can be imagined as the length needed to move between two points in a grid where you can only move up, down, left or right.

Extension

This definition can be used for three and higher dimensions too. If there are two vectors, $\mathbf {p} =(p_{1},p_{2}\dots ,p_{n})$ and $\mathbf {q} =(q_{1},q_{2}\dots ,q_{n})$, then the manhattan distance between the two points is the absolute value of the difference between all numbers in the vector. Or, in notation: $$\sum _{i=1}^{n}|p_{i}-q_{i}|$$