tags:
  - math
  - statistics
date: 2025-07-31
aliases:
  - City Block Distance

Manhattan Distance

Author: Ernad Mujakic
Date: 2025-07-31

The Manhattan distance is a Distance Measure defined as the sum of the absolute differences of the Cartesian coordinates of two points. It is the distance between two points using only grid-like movements (horizontal and vertical). The Manhattan distance is also the norm of the distance between two vectors in Lp Space.

The Manhattan distance between two objects in -dimensional space is calculated as:

Properties

Positive: Like any other distance metric, the range of the Manhattan distance is where a 0 distance indicates that the two points are at the same location.
Symmetric: The Manhattan distance is symmetric, meaning
Triangle Inequality: The Manhattan distance obeys the triangle inequality, which states that the distance from to is always less than or equal to the distance from to plus the distance from to . Meaning that taking a detour through a third point cannot result in a shorter distance than a direct path from to .

Applications

Machine Learning

Clustering: Manhattan distance may be used in clustering algorithms like K-Means Clustering to group similar data points.
Classification: The Manhattan distance may be employed in distance-based classification algorithms such as K-Nearest-Neighbors.

Computer Vision

Measuring Pixel Differences: The Manhattan distance, or other distance measures, may be employed to compare pixel values, or features in image recognition tasks.

Pathfinding

The Manhattan distance can be used for calculating the shortest path between points. It is also an Admissible and Consistent Heuristic in search algorithms such as A* Search.
Grid-Navigation: The Manhattan distance is particularly useful for pathfinding in domains which only allow grid-like movements, such as a chessboard or a city network.

Statistics

Outlier Detection: Manhattan distance can be used for distance based outlier detection algorithms such as DBSCAN.
Multidimensional Scaling: Manhattan distance may be employed to visualize higher-dimensional data into lower dimensions.

Other Distance Measures

Other common distance measures include:

Euclidean Distance: The straight-line distance between 2 points in Euclidean space.
Chebyshev Distance: The maximum absolute difference between 2 vectors across all dimensions.
Minkowski Distance: A generalized distance measure that is defined by a parameter whose common values are the Manhattan distance, Euclidean distance, and Chebyshev distance.
Jaccard Index: Used to compare sets and is defined as the size of the intersection of 2 sets, over the size of their union.

References

J. Han and M. Kamber, Data Mining : Concepts and Techniques, 3rd ed. Haryana, India ; Burlington, Ma: Elsevier, 2018.
“Taxicab geometry,” Wikipedia, Jan. 21, 2022. https://en.wikipedia.org/wiki/Taxicab_geometry
GeeksforGeeks, “Clustering Distance Measures,” GeeksforGeeks, May 24, 2024. https://www.geeksforgeeks.org/machine-learning/clustering-distance-measures/#common-distance-measures (accessed Jul. 31, 2025).