WOLFRAM

Wolfram Language & System Documentation Center

ManhattanDistance[u,v]

gives the Manhattan or "city block" distance between vectors u and v.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
See Also
Tech Notes
Related Guides
History
Cite this Page

ManhattanDistance

ManhattanDistance[u,v]

gives the Manhattan or "city block" distance between vectors u and v.

Details

  • Manhattan distance is effectively the sum of of differences across all dimensions.
  • ManhattanDistance[u,v] is equivalent to Total[Abs[u-v]]. »

Examples

open all close all

Basic Examples  (2)

Manhattan distance between two vectors:

Wolfram Language code: ManhattanDistance[{a, b, c}, {x, y, z}]

Manhattan distance between numeric vectors:

Wolfram Language code: ManhattanDistance[{1, 2, 3}, {2, 4, 6}]

Scope  (2)

Compute distance between any vectors of equal length:

Wolfram Language code: ManhattanDistance[RandomReal[5, 100], RandomReal[5, 100]]

Compute distance between vectors of any precision:

Wolfram Language code: ManhattanDistance[N[{1, 5, 2, 3, 10}, 50], N[{4, 15, 20, 5, 5}, 50]]

Applications  (2)

Cluster data using Manhattan distance:

Wolfram Language code: FindClusters[{{2, 3}, {5, 10}, {4, 5}, {2, 2}}, DistanceFunction -> ManhattanDistance]

Demonstrate the triangle inequality:

Wolfram Language code: d1 = ManhattanDistance[{a, b}, {a, c}]
Wolfram Language code: d2 = ManhattanDistance[{a, c}, {d, c}]
Wolfram Language code: d3 = ManhattanDistance[{a, b}, {d, c}]
Wolfram Language code: Simplify[d3 <= d1 + d2]

Properties & Relations  (5)

Manhattan distance is a sum of absolute differences:

Wolfram Language code: ManhattanDistance[{a, b, c}, {x, y, z}]
Wolfram Language code: Total[Abs[{a, b, c} - {x, y, z}]]

ManhattanDistance is equivalent to a Norm of a difference:

Wolfram Language code: ManhattanDistance[{a, b, c}, {x, y, z}]
Wolfram Language code: Norm[{a, b, c} - {x, y, z}, 1]

ManhattanDistance is greater than or equal to ChessboardDistance:

Wolfram Language code: u = {a, b, c}; v = {x, y, z};
Wolfram Language code: Simplify[ManhattanDistance[u, v] ≥ ChessboardDistance[u, v]]

BrayCurtisDistance is a ratio of Manhattan distances:

Wolfram Language code: u = {a, b, c}; v = {x, y, z};
Wolfram Language code: ManhattanDistance[u, v] / ManhattanDistance[u, -v]
Wolfram Language code: BrayCurtisDistance[u, v]

MeanDeviation as a scaled ManhattanDistance from the Mean:

Wolfram Language code: data = {a, b, c}
Wolfram Language code: mean = Table[Mean[data], {Length[data]}]
Wolfram Language code: MeanDeviation[data]
Wolfram Language code: ManhattanDistance[data, mean] / Length[data]
Wolfram Research (2007), ManhattanDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/ManhattanDistance.html.

Text

Wolfram Research (2007), ManhattanDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/ManhattanDistance.html.

CMS

Wolfram Language. 2007. "ManhattanDistance." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ManhattanDistance.html.

APA

Wolfram Language. (2007). ManhattanDistance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ManhattanDistance.html

BibTeX

@misc{reference.wolfram_2026_manhattandistance, author="Wolfram Research", title="{ManhattanDistance}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/ManhattanDistance.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_manhattandistance, organization={Wolfram Research}, title={ManhattanDistance}, year={2007}, url={https://reference.wolfram.com/language/ref/ManhattanDistance.html}, note=[Accessed: 12-June-2026]}