Manhattan Distance

Overview

Manhattan distance (L1 norm) calculates the distance between vectors by summing the absolute differences of their components. Also known as taxicab or city block distance.

Formula

Distance = Σ|a[i] - b[i]|

Characteristics

• Grid Distance: Measures path along axes (like Manhattan city blocks)
• Outlier Robust: Less sensitive to outliers than Euclidean
• Fast Computation: Values typically smaller than Euclidean
• High-Dimensional Friendly: Better performance as dimensions increase

When to Use

• High-dimensional data
• When outliers are present
• Computational efficiency is important
• Grid-like or discrete data

Performance Benefits

• Faster to calculate than Euclidean distance
• Values are typically smaller
• Recommended as dimensionality increases

Comparison

• Manhattan: Grid distance, robust to outliers
• Euclidean: Straight-line distance, sensitive to outliers
• Cosine: Angular similarity, scale invariant

Use Cases

• High-dimensional feature spaces
• Data with potential outliers
• Performance-sensitive applications
• Discrete or grid-structured data

Limitations

Less intuitive than Euclidean distance for geometric problems. May not reflect actual similarity for normalized or angular comparisons.

Vector Database Support

Supported by some vector databases, though less common than cosine and Euclidean. Check specific database documentation.