Manhattanavståndet

Manhattanavståndet, also known as the taxicab metric or L1 distance, is a metric defined on a vector space where the distance between two points is the sum of the absolute differences of their Cartesian coordinates. It is named after the grid layout of the streets on Manhattan Island, where a taxi or pedestrian must travel along the grid lines to get from one point to another, rather than a straight line.

Formally, for two points p = (p1, p2, ..., pn) and q = (q1, q2, ..., qn) in an n-dimensional space, the Manhattan distance dp(p, q) is calculated as:

dp(p, q) = |p1 - q1| + |p2 - q2| + ... + |pn - qn|

This can also be expressed using summation notation as:

dp(p, q) = Σ (from i=1 to n) |pi - qi|

The Manhattan distance is a common distance metric in various fields, including urban planning, computer science (especially in image processing and machine learning for tasks like clustering and classification), and game development. Unlike the Euclidean distance (which represents the straight-line distance), the Manhattan distance reflects movement constrained to orthogonal directions. For example, in a city grid, travelling 1 block north and then 1 block east covers a Manhattan distance of 2 blocks, even though the direct Euclidean distance would be approximately 1.4 blocks (the square root of 2). It is a specific case of the Lp metric where p=1.