Manhattandistance

The Manhattan distance, also known as the L1 distance, taxicab distance, or city block distance, is a metric in a vector space where the distance between two points is the sum of the absolute differences of their coordinates. Imagine navigating a city grid like Manhattan, where you can only travel along perpendicular streets. The Manhattan distance represents the shortest path you could take between two points under these constraints.

Mathematically, for two points $p = (p_1, p_2, ..., p_n)$ and $q = (q_1, q_2, ..., q_n)$ in an n-dimensional vector space, the Manhattan distance is defined as: $d_1(p, q) = \sum_{i=1}^{n} |p_i - q_i|$.

This metric is distinct from the more commonly known Euclidean distance (L2 distance), which calculates the straight-line distance between two points. In a two-dimensional plane, the Euclidean distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$. In contrast, the Manhattan distance would be $|x_1 - x_2| + |y_1 - y_2|$.

The Manhattan distance finds applications in various fields. In computer science, it's used in areas like image processing, machine learning (e.g., in clustering algorithms and feature selection), and robotics. It's particularly useful when movement is restricted to orthogonal directions, such as in certain types of games or in analyzing grid-based systems. Its computational simplicity also makes it an attractive choice in some applications.