What is Chebyshev distance and how is it calculated?

Chebyshev distance is the largest absolute difference across any single coordinate of two equal-length vectors: d∞ = maxᵢ |aᵢ − bᵢ|. Subtract the vectors coordinate by coordinate, take the absolute value of each difference, and keep the biggest one. For a = (2, 3, 5) and b = (7, 1, 6) the absolute differences are 5, 2 and 1, so the Chebyshev distance is 5.

What is the difference between Chebyshev, Euclidean and Manhattan distance?

All three measure how far apart two points are but weight the coordinate differences differently. Manhattan (L1) adds every absolute difference, Euclidean (L2) takes the square root of the summed squares, and Chebyshev (L∞) keeps only the single largest difference. For the same pair the order is always Chebyshev ≤ Euclidean ≤ Manhattan, because one term can never exceed a sum that includes it.

Why is Chebyshev distance called the L-infinity or chessboard distance?

It is the p → ∞ member of the Minkowski (Lₚ) family, where the largest term dominates the sum, hence L∞. It is the chessboard distance because on a grid it equals the minimum king moves between two squares: a king steps one square in any of eight directions, so it needs max(|Δfile|, |Δrank|) moves. Moving from e1 to h5 is max(3, 4) = 4 moves.

When should I use Chebyshev distance in machine learning or KNN?

Use it when a single most-divergent feature should decide nearness, or when movement is grid-based with diagonal steps allowed at the same cost as straight ones — warehouse robots, tile maps, and some image-dilation operations. In k-nearest-neighbours, scikit-learn exposes it as metric "chebyshev". It is less sensitive to many small differences than Euclidean and more sensitive to one large outlier dimension.

Is Chebyshev distance a special case of Minkowski distance?

Yes. The Minkowski distance of order p is (Σ |aᵢ − bᵢ|ᵖ)^(1/p). As p grows, the largest term increasingly dominates the sum, and in the limit p → ∞ the whole expression collapses to maxᵢ |aᵢ − bᵢ| — exactly the Chebyshev distance. This tool shows the Minkowski distance at p = 3 alongside Chebyshev so you can watch the metrics tighten toward the maximum.

What does the dominant dimension mean?

The dominant dimension is the coordinate that produced the maximum absolute difference — the one that actually sets the Chebyshev distance. The tool highlights it in the working table with a crown. When two or more dimensions tie for the maximum it reports the count, which matters because ties are why the Minkowski approximation converges to Chebyshev more slowly.

Why must the two vectors have the same number of components?

The metric pairs coordinates by position — a₁ with b₁, a₂ with b₂, and so on — so both vectors need the same length for the pairing to be defined. Comparing a 2-D point with a 3-D point has no meaning. If the lengths differ, the calculator stops and names both lengths instead of returning a misleading number.

Does this calculator match SciPy and scikit-learn, and is my data sent anywhere?

The result equals scipy.spatial.distance.chebyshev(u, v) and scikit-learn's DistanceMetric "chebyshev". Nothing is uploaded: parsing your numbers, taking the differences and finding the maximum all run in your browser with plain JavaScript, so you can paste real feature vectors without concern. The tool also recomputes the answer a second way, by sorting, and confirms the two agree.

AI · Machine learning

Chebyshev Distance Calculator

Compute the Chebyshev (L∞) distance between two points or two vectors of any dimension, in your browser. It is the largest single coordinate difference — the chessboard king-move metric — shown with the dominant dimension and the Manhattan, Euclidean and Minkowski comparisons. No signup, nothing uploaded.

By Induwara Ashinsana— Executive Director, Ryzera TechnologiesUpdated Jul 16, 2026

Chebyshev distance calculator

Input mode

Vector / Point A

Any length ≥ 1 — A and B must match.

Vector / Point B

Same length as A. Commas, spaces, or new lines.

Examples

Chebyshev distance

5.0000

L∞ · 3-D

Minkowski (p=3)

5.1172

(Σ |aᵢ − bᵢ|³)^⅓

Euclidean (L2)

5.4772

√Σ (aᵢ − bᵢ)²

Manhattan (L1)

8.0000

Σ |aᵢ − bᵢ|

Dominant dimension. Dimension #1 sets the Chebyshev distance with the largest absolute difference of 5.0000.

Ordering holds: Chebyshev ≤ Minkowski(p=3) ≤ Euclidean ≤ Manhattan (5.0000 ≤ 5.1172 ≤ 5.4772 ≤ 8.0000).

Decimals

Cross-check. The single-pass running maximum gives 5.0000; an independent route — sorting the absolute differences and taking the largest — gives 5.0000. They match exactly, as they must.

Step-by-step working

Dim	aᵢ	bᵢ	aᵢ − bᵢ	\|aᵢ − bᵢ\|
#1	2.0000	7.0000	-5.0000	5.0000
#2	3.0000	1.0000	2.0000	2.0000
#3	5.0000	6.0000	-1.0000	1.0000
Totals			Σ\|·\| = 8.0000	max = 5.0000

d∞ = max(|−5.0000|, |2.0000|, |−1.0000|) = 5.0000

How it works

Chebyshev distance — also called the L∞, maximum, or chessboard distance — is the largest of the coordinate-wise absolute differences between two points. For two equal-length vectors a = [a₁…aₙ] and b = [b₁…bₙ], only the single most-divergent dimension matters. It is the metric SciPy exposes as distance.chebyshev and scikit-learn exposes as the KNN metric "chebyshev".

d∞(a, b) = maxᵢ |aᵢ − bᵢ| = max( |a₁ − b₁|, … , |aₙ − bₙ| )

The tool computes this in three steps:

Per-dimension difference. Subtract the vectors coordinate by coordinate and take the absolute value: |dᵢ| = |aᵢ − bᵢ|.
Take the maximum. The Chebyshev distance is the largest of those absolute differences, max |dᵢ|. The dimension that attains it is the dominant dimension; when several tie, they all count.
Comparison metrics. Over the same differences the tool also reports Manhattan Σ |dᵢ|, Euclidean √Σ dᵢ² and Minkowski of order 3 (Σ |dᵢ|³)^⅓, the other members of the same Lₚ family.

Chebyshev is the p → ∞ limit of the Minkowski distance. Writing the Minkowski distance as max·(Σ (|dᵢ|/max)^p)^(1/p), every ratio below 1 vanishes as p grows and only the maximal term survives, so the whole expression tends to the maximum. That is why the ordering Chebyshev ≤ Minkowski(3) ≤ Euclidean ≤ Manhattan always holds for the same pair of points, and the calculator surfaces it under the result. All arithmetic is done in double-precision floating point and rounded only for display; the Minkowski route normalises by the maximum first, so even inputs around 10⁹ never overflow. As a credibility check the tool recomputes the Chebyshev distance a second way — by sorting the absolute differences and taking the largest — and confirms the two routes agree exactly.

Worked examples

3-D feature vectors — a = [2, 3, 5], b = [7, 1, 6]

Absolute differences: |2 − 7| = 5, |3 − 1| = 2, |5 − 6| = 1
Chebyshev: max(5, 2, 1) = 5 (dominant: dimension #1)
Manhattan: 5 + 2 + 1 = 8, Euclidean: √30 = 5.477226
Minkowski(3): (125 + 8 + 1)^⅓ = 134^⅓ = 5.117312
Ordering 5 ≤ 5.117 ≤ 5.477 ≤ 8 → holds, scipy.chebyshev = 5

Chess king moves — e1 → h5, mapped to a = [5, 1], b = [8, 5]

Files a–h map to 1–8, so e1 = (5, 1) and h5 = (8, 5)
Absolute differences: |5 − 8| = 3, |1 − 5| = 4
Chebyshev: max(3, 4) = 4 → the king needs 4 moves
Manhattan: 7, Euclidean: 5, Minkowski(3): 91^⅓ = 4.497941
Matches the standard chessboard-distance result

Tie across dimensions — p = [1.5, −2], q = [4.5, 1]

Absolute differences: |1.5 − 4.5| = 3.0, |−2 − 1| = 3.0
Chebyshev: max(3.0, 3.0) = 3.0 (2 dimensions tie)
Manhattan: 6.0, Euclidean: √18 = 4.242641
Minkowski(3): 54^⅓ = 3.779763
Ordering 3.0 ≤ 3.780 ≤ 4.243 ≤ 6.0 → holds

Frequently asked questions

Sources & references

The formulas on this page were last cross-checked against these sources on 2026-07-16. Chebyshev distance is a stable mathematical definition, so this tool needs no rate or schedule updates — only the worked examples are periodically re-reconciled.

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