Manhattan Distance

Synonyms

City block distance; L₁-distance; 1-norm distance; Taxicab norm distance

Definition

The Manhattan distance between two points x = (x₁, x₂, … x _n ) and y = (y₁, y₂, … y _n ) in n-dimensional space is the sum of the distances in each dimension:

$$\displaystyle{d(\mathbf{x,y}) =\sum _{ i=1}^{n}\mid x_{ i} - y_{i}\mid }$$

It is called the Manhattan distance because it is the distance a car would drive in a city (e.g., Manhattan) where the buildings are laid out in square blocks and the straight streets intersect at right angles. This explains the other terms city block and taxicab distances. The terms L₁ and 1-norm distances are the mathematical descriptions of this distance.

Cross-References

• Case-Based Reasoning

• Nearest Neighbor