Minkowski Distance Calculator

The Minkowski distance is the generalized Lₚ metric. For two equal-length vectors a = [a₁…aₙ] and b = [b₁…bₙ] and an order p ≥ 1, it is the p-th root of the sum of the p-th powers of the absolute coordinate differences. It is the metric family scikit-learn and SciPy use for nearest-neighbour search, and Manhattan, Euclidean, and Chebyshev distance are all special cases of it.

dₚ(a, b) = ( Σ |aᵢ − bᵢ|ᵖ )^(1/p) = ( |a₁ − b₁|ᵖ + … + |aₙ − bₙ|ᵖ )^(1/p)

The tool computes this in three steps:

1. Per-dimension absolute difference. Subtract the vectors coordinate by coordinate and take the magnitude: dᵢ = |aᵢ − bᵢ| (the NIST / SciPy definition).
2. Raise to p and sum. Raise each difference to the power p and add them: S = Σ dᵢᵖ — the inner sum used by scikit-learn's DistanceMetric "minkowski".
3. Take the p-th root. The distance is dₚ = S^(1/p), matching scipy.spatial.distance.minkowski(a, b, p).

The named metrics are just specific orders of the same sum, and the tool reports all three alongside your chosen p:

• p = 1 → Manhattan (L1): Σ |aᵢ − bᵢ| — the city-block path.
• p = 2 → Euclidean (L2): √( Σ (aᵢ − bᵢ)² ) — the straight line.
• p → ∞ → Chebyshev (L∞): maxᵢ |aᵢ − bᵢ| — the largest single-axis gap.

Chebyshev is computed directly via the maximum rather than by plugging a huge number in for p, because |aᵢ − bᵢ|ᵖ would overflow to infinity for a large finite p. p ≥ 1is enforced because the triangle inequality — and therefore a true metric — fails for 0 < p < 1. All arithmetic is done in double-precision floating point on the raw inputs and is rounded only for display. As a credibility check, the calculator recomputes the distance a second way — an overflow-safe scaled form that factors out the largest difference before raising it to p — and confirms the two routes agree, so even inputs around 10⁹ stay exact.

• scikit-learn — sklearn.metrics.DistanceMetric, "minkowski" (definition d(x,y) = (Σ |xᵢ − yᵢ|ᵖ)^(1/p); default KNN metric at p=2)
• SciPy — scipy.spatial.distance.minkowski (reference implementation and the p ≥ 1 validity constraint; the cross-check oracle)
• NIST Dictionary of Algorithms and Data Structures — Lₘ distance / Minkowski metric and its limiting cases (Manhattan, Euclidean, Chebyshev)

The formulas on this page were last cross-checked against these sources on 2026-06-11. Minkowski distance is a stable mathematical definition, so this tool needs no rate or schedule updates — only the worked examples are periodically re-reconciled against SciPy.