ManhattanAbstand

ManhattanAbstand, commonly known as the Manhattan distance or L1 distance, is a metric used to measure the separation between two points in grid-like spaces. For points x = (x1, ..., xn) and y = (y1, ..., yn) in R^n, the ManhattanAbstand is defined as d1(x,y) = Σ_i |xi - yi|. It is the L1 norm of the difference x - y, written ||x - y||1. The name originates from the grid-like street pattern of Manhattan, New York, where travel is often along axis-aligned blocks.

In two dimensions, the distance equals the number of axis-aligned steps required to move from x to y on a grid: d1(x,y) = |x1 - y1| + |x2 - y2|. In higher dimensions, it generalizes by summing the absolute coordinate differences.

ManhattanAbstand satisfies all the properties of a metric: non-negativity, zero distance only when the points coincide, symmetry, and the triangle inequality. It scales linearly with a common factor: d1(ax, ay) = |a| d1(x,y).

Geometrically, the unit ball of the L1 norm in two dimensions is a diamond (a square rotated 45 degrees). This differs from the Euclidean metric, whose unit circle is a circle, reflecting different notions of distance.

Applications include path planning on grid graphs, clustering methods such as k-medians, nearest-neighbor search in high-dimensional data, and error measurement in digital imagery where axis-aligned differences are natural. ManhattanAbstand remains a fundamental alternative to the Euclidean distance in contexts with grid-like structure or axis-aligned movement.