Manhattanetäisyys

Manhattanetäisyys, also known as Manhattan distance or L1 distance, is a metric used to measure the distance between two points in a multi-dimensional space. The term originates from the idea of navigating a city grid, such as Manhattan, where movement is restricted to horizontal and vertical directions rather than diagonal paths. In this context, the distance between two points is calculated as the sum of the absolute differences of their Cartesian coordinates.

Mathematically, for two points in an n-dimensional space, denoted as (x₁, x₂, ..., xₙ) and (y₁, y₂, ..., yₙ), the Manhattan distance is defined as the sum of the absolute differences of their corresponding coordinates:

d((x₁, x₂, ..., xₙ), (y₁, y₂, ..., yₙ)) = |x₁ - y₁| + |x₂ - y₂| + ... + |xₙ - yₙ|

This metric is particularly useful in applications where movement is constrained to axes-aligned directions, such as in grid-based pathfinding or certain types of image processing. Unlike the Euclidean distance, which measures the straight-line distance between two points, Manhattan distance does not account for diagonal movement, making it a simpler and often more practical choice in specific scenarios.

Manhattanetäisyys is also employed in various fields such as computer science, statistics, and operations research. For instance, it is used in k-nearest neighbors algorithms for classification and regression tasks, where the similarity between data points is measured using this distance metric. Additionally, it plays a role in optimization problems, particularly those involving the traveling salesman problem or network routing.

One notable property of Manhattan distance is its robustness to outliers, as it is less sensitive to extreme values compared to Euclidean distance. This characteristic makes it a preferred choice in certain data analysis applications where data may contain noise or anomalies.