3. Manhattan Distance and Other Metrics

Overview

Besides Euclidean, there are other distance metrics. The best known is Manhattan (a path along a grid).

---

Manhattan distance

Formula:

manhattan_distance(A, B) = |b₁ - a₁| + |b₂ - a₂| + ... + |bₙ - aₙ|
                         = ∑|bᵢ - aᵢ|

2D example:

A = [2, 3]
B = [5, 7]

Manhattan = |5-2| + |7-3| = 3 + 4 = 7

Interpretation: The distance you walk along a grid (city streets).

---

Visual comparison (2D)

        ↑ B(5,7)
        |  ╱|
      7 | ╱ |
        |╱  | ← Manhattan (the grid path)
      3 A───→
        2   5

Euclidean (diagonal): 5
Manhattan (grid): 7

---

Other metrics

Minkowski distance (the generalization):

distance = (∑|bᵢ - aᵢ|ᵖ)^(1/p)

p=1: Manhattan
p=2: Euclidean
p=∞: Chebyshev (the largest difference on any single axis)

---

When to use Manhattan

Use cases:

• Movement on a grid (robots in a warehouse)
• Categorical data (differences across attributes)
• When outliers are a problem (Manhattan is more robust)

In AI: Rare for embeddings (Euclidean or cosine are more common).

---

Next capsule: 04-cosine-similarity.md — The key metric for AI.