Manhattan Distance Calculator

What is Manhattan Distance (Taxicab Metric)?

The Manhattan distance, also known as the taxicab distance, city block distance, or $L_1$ norm, is the distance between two points measured along axes at right angles. The name originates from the grid layout of streets on the island of Manhattan in New York City, where a vehicle must travel along perpendicular avenues and streets rather than cutting diagonally through buildings.

In mathematics, physics, and computer science, the Manhattan distance between two points in an $n$-dimensional Cartesian coordinate space is defined as the sum of the absolute differences of their Cartesian coordinates:

$$d_1(P, Q) = \|P - Q\|_1 = \sum_{i=1}^n |p_i - q_i|$$

For direct straight-line distance calculations in 2D or 3D spaces, you can also explore our 2D Distance Calculator, 3D Distance Calculator, and Cosine Similarity Calculator.

Manhattan Distance Formulas Across Dimensions

1. In Two Dimensions (2D)

For two points $A = (x_1, y_1)$ and $B = (x_2, y_2)$:

$$d_1(A, B) = |x_1 - x_2| + |y_1 - y_2|$$

2. In Three Dimensions (3D)

For two points $A = (x_1, y_1, z_1)$ and $B = (x_2, y_2, z_2)$:

$$d_1(A, B) = |x_1 - x_2| + |y_1 - y_2| + |z_1 - z_2|$$

3. In Four Dimensions (4D)

For two points $A = (x_1, y_1, z_1, w_1)$ and $B = (x_2, y_2, z_2, w_2)$:

$$d_1(A, B) = |x_1 - x_2| + |y_1 - y_2| + |z_1 - z_2| + |w_1 - w_2|$$

Manhattan vs. Euclidean vs. Chebyshev Distance

Metric	Norm Notation	Formula in 2D	Physical Interpretation
Manhattan Distance	$L_1$ Norm	$\|x_1 - x_2\| + \|y_1 - y_2\|$	Movement strictly restricted to orthogonal grid streets (taxi navigation).
Euclidean Distance	$L_2$ Norm	$\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$	Straight line distance ("as the crow flies").
Chebyshev Distance	$L_\infty$ Norm	$\max(\|x_1 - x_2\|, \|y_1 - y_2\|)$	King moves on a chessboard (diagonal steps cost the same as orthogonal).

Because the straight line represents the shortest path between two points in flat Euclidean geometry, the inequality always holds:

$$d_\infty(A, B) \le d_2(A, B) \le d_1(A, B) \le \sqrt{n} \cdot d_2(A, B)$$

Worked Step-by-Step Example

Calculate the Manhattan distance between Point $A = (1, 2, 5)$ and Point $B = (8, 9, 1)$:

Compute difference on the x-axis: $|1 - 8| = |-7| = 7$.
Compute difference on the y-axis: $|2 - 9| = |-7| = 7$.
Compute difference on the z-axis: $|5 - 1| = |4| = 4$.
Add all differences: $d_1 = 7 + 7 + 4 = 18$.
Compare to Euclidean distance: $d_2 = \sqrt{7^2 + 7^2 + 4^2} = \sqrt{49 + 49 + 16} = \sqrt{114} \approx 10.6771$.

Applications of Manhattan Distance

Machine Learning and Data Science: Used as the distance metric in $k$-Nearest Neighbors ($k$-NN) and clustering algorithms when dealing with high-dimensional data or features with non-continuous attributes.
Robotics and Pathfinding: The standard heuristic function in the $A^*$ search algorithm for 4-directional grid maps.
Integrated Circuit (IC) Layout: Routing metal interconnect wires in microchips that run exclusively horizontally and vertically across semiconductor layers.
Urban Logistics and Supply Chain: Estimating delivery routes and taxi travel times in city centers laid out on rectangular street grids.

Frequently Asked Questions

Can Manhattan distance be negative?

No. Because Manhattan distance is defined as the sum of absolute values $|a_i - b_i|$, every term is non-negative ($\ge 0$). Thus, the Manhattan distance is always greater than or equal to zero, and is zero if and only if Point A and Point B are identical.

When should I use Manhattan distance instead of Euclidean distance?

Use Manhattan distance when movement is restricted to a grid (such as city streets or grid-based video games), when dealing with sparse or high-dimensional data where the $L_1$ norm is less sensitive to extreme outliers, or when calculating $A^*$ heuristics for 4-directional movement.

Is there only one shortest Manhattan path between two points?

No. While the numerical distance value is unique, there are often multiple distinct paths on a grid that share the exact same minimal Manhattan length. Any monotonic path that only moves in the direction of the target without backtracking has the same total distance.

What is the relationship between Manhattan distance and Minkowski distance?

The Minkowski distance is a generalized metric defined as $D(X, Y) = (\sum |x_i - y_i|^p)^{1/p}$. When $p = 1$, the Minkowski distance simplifies exactly to the Manhattan distance ($L_1$ norm). When $p = 2$, it becomes Euclidean distance ($L_2$ norm).

Report