## Definition Edit

Given a number set $E$, the Manhattan distance is a function $ManhattanDis:E^n\times E^n \longrightarrow \mathbb{R}$ defined as

$ManhattanDis(u,v)=\sum_{i=1}^{n}|u[i]-v[i]|.$

## Examples Edit

• $ManhattanDis((1,2,3,4),(4,3,2,1)) = 3+1+1+3 = 8$.
• $ManhattanDis((1,0,1,0,1),(1,1,0,0,1)) = 0+1+1+0+0 = 2$.

## Normalization Edit

If $E$ is a bounded set, is possible to normalize the difference dividing by the range of $E$, then normalization is

$ManhattanSim(u,v)=1-\frac{\sum_{i=1}^{n}|u[i]-v[i]|}{n}.$

that is the aritmetic mean of the normalized differences.

## Examples Edit

• If $E=[0,10]$, $ManhattanSim((1,2,3,4),(4,3,2,1)) = 1-\frac{0.3+0.1+0.1+0.3}{4} = 0.8$.
• If $E=\{0,1\}$, $ManhattanSim((1,0,1,0,1),(1,1,0,0,1)) = 1-\frac{0+1+1+0+0}{5} = 0.6$.

## Variations Edit

• Manhattan distance is a particular case of Minkowski distance when $p=1$.
• If $E=\{0,1\}$, whe have the Hamming similarity.

## Applications Edit

• Numeric vectors (codes).
• Vectors of boolean features.