Manhattan distance (L1)

The Manhattan distance, also called taxicab distance or the L1 norm, measures distance in a grid-like space: the sum of the absolute differences of the coordinates.

In n dimensions, for A(a₁, …, aₙ) and B(b₁, …, bₙ):

Worked example: $A(3,5)$ and $B(6,9)$ give $d(A,B)=|6-3|+|9-5|=3+4=7$.

Like every metric it is non-negative, zero only when A = B, symmetric, and satisfies the triangle inequality d(A,B) ≤ d(A,C) + d(C,B).

It is used for grid-based pathfinding (games, city-block navigation), in clustering and classification algorithms such as k-means and k-NN, in optimisation where movement is restricted to orthogonal directions, and in robotics.