Manhattan distance

Definition

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

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

Normalization

If $E$ is a bounded set, it 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 arithmetic mean of the normalized differences.

Examples

• 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

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

Applications

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

References

• http://en.wikipedia.org/wiki/Manhattan_distance