Manhattan Distance Calculator
Calculate L1 distance (taxicab distance) between points in 2D and 3D space
Manhattan Distance Calculator
Calculate the sum of absolute differences between coordinates
2D Visualization
Red dashed line shows Manhattan distance path (right-angle turns only)
Understanding Manhattan Distance
Manhattan distance, also known as taxicab distance, city block distance, or L1 distance, is a distance metric that measures the distance between two points as the sum of the absolute differences of their coordinates. Unlike Euclidean distance which measures the straight-line distance, Manhattan distance represents the distance a taxi would travel in a city laid out in a grid pattern, where movement is restricted to horizontal and vertical directions only.
Distance Formulas
2D Manhattan Distance
d = |x₂ - x₁| + |y₂ - y₁|
Sum of absolute differences in each dimension
3D Manhattan Distance
d = |x₂ - x₁| + |y₂ - y₁| + |z₂ - z₁|
Extended to three dimensions with z-coordinate
Mathematical Properties
Metric Properties
- ✓Non-negativity: d(x, y) ≥ 0
- ✓Identity: d(x, y) = 0 if and only if x = y
- ✓Symmetry: d(x, y) = d(y, x)
- ✓Triangle inequality: d(x, z) ≤ d(x, y) + d(y, z)
Key Characteristics
- •Always results in integer values for integer coordinates
- •Generally larger than Euclidean distance for the same points
- •Computationally simpler (no square root calculation)
- •Forms diamond-shaped unit circles in 2D space
Distance Metrics Comparison
| Metric | Formula (2D) | Use Case | Example Result |
|---|---|---|---|
| Manhattan | |x₂-x₁| + |y₂-y₁| | Grid navigation, city blocks | (0,0) to (3,4) = 7 |
| Euclidean | √[(x₂-x₁)² + (y₂-y₁)²] | Straight-line distance | (0,0) to (3,4) = 5 |
| Chebyshev | max(|x₂-x₁|, |y₂-y₁|) | Chess king moves | (0,0) to (3,4) = 4 |
Real-World Applications
Urban Planning
- • City block distance calculations
- • Taxi route optimization
- • Emergency service coverage
- • Public transport planning
Computer Science
- • Machine learning algorithms
- • Image processing
- • Data clustering
- • Pathfinding in grids
Logistics
- • Warehouse navigation
- • Delivery route planning
- • Supply chain optimization
- • Inventory placement
Gaming & Robotics
- • Grid-based game movement
- • Robot navigation
- • Turn-based strategy games
- • Tile-based pathfinding
Data Analysis
- • Feature similarity measurement
- • Outlier detection
- • Market basket analysis
- • Recommendation systems
Operations Research
- • Facility location problems
- • Resource allocation
- • Network analysis
- • Optimization problems
Worked Examples
2D Example: City Block Navigation
Problem:
Find the Manhattan distance from corner (0, 0) to building at (3, 4)
Point A: (0, 0)
Point B: (3, 4)
Δx = |3 - 0| = 3
Δy = |4 - 0| = 4
Manhattan Distance = 3 + 4 = 7 blocks
This means you need to walk 7 city blocks total (3 east + 4 north)
3D Example: Warehouse Navigation
Problem:
Distance from storage (1, 2, 0) to shelf (4, 6, 3)
Point A: (1, 2, 0)
Point B: (4, 6, 3)
Δx = |4 - 1| = 3
Δy = |6 - 2| = 4
Δz = |3 - 0| = 3
Manhattan Distance = 3 + 4 + 3 = 10 units
Total movement: 3 units along x-axis, 4 along y-axis, 3 along z-axis
Algorithm Implementation
Python Implementation
def manhattan_distance_2d(x1, y1, x2, y2):
return abs(x2 - x1) + abs(y2 - y1)
def manhattan_distance_3d(x1, y1, z1, x2, y2, z2):
return abs(x2 - x1) + abs(y2 - y1) + abs(z2 - z1)
# Example usage
dist_2d = manhattan_distance_2d(0, 0, 3, 4) # Result: 7
dist_3d = manhattan_distance_3d(1, 2, 0, 4, 6, 3) # Result: 10JavaScript Implementation
function manhattanDistance2D(x1, y1, x2, y2) {
return Math.abs(x2 - x1) + Math.abs(y2 - y1);
}
function manhattanDistance3D(x1, y1, z1, x2, y2, z2) {
return Math.abs(x2 - x1) + Math.abs(y2 - y1) + Math.abs(z2 - z1);
}
// Example usage
const dist2D = manhattanDistance2D(0, 0, 3, 4); // Result: 7
const dist3D = manhattanDistance3D(1, 2, 0, 4, 6, 3); // Result: 10Advanced Concepts
Weighted Manhattan Distance
Different dimensions can be given different importance using weights:
d = w₁|x₂-x₁| + w₂|y₂-y₁| + w₃|z₂-z₁|
Useful when some dimensions are more important than others in the analysis.
Fractional Manhattan Distance
Generalization to fractional powers (Minkowski distance with p=1):
d = (Σ|xᵢ₂ - xᵢ₁|ᵖ)^(1/p) where p = 1
Manhattan distance is a special case of Minkowski distance with p=1.
Computational Complexity
Manhattan distance has O(n) time complexity for n-dimensional points, making it computationally efficient for high-dimensional data analysis. Unlike Euclidean distance, it doesn’t require expensive square root calculations.
Advantages and Limitations
Advantages
- • Computationally efficient (no square roots)
- • Intuitive for grid-based problems
- • Works well with categorical data
- • Less sensitive to outliers than Euclidean distance
- • Natural for many real-world scenarios
- • Always produces integer results for integer inputs
Limitations
- • Doesn’t represent true physical distance
- • May overestimate distances in free space
- • Not suitable for continuous geometric problems
- • Can be biased by coordinate system orientation
- • May not capture diagonal relationships well
- • Less meaningful for non-grid environments