City Block distance and Chessboard distance are two fundamental metrics used to calculate the distance between two points in a grid-like space, commonly encountered in fields like image processing, robotics, and game development. They represent different ways of navigating from one point to another, based on allowed movements.
City Block Distance (Manhattan Distance)
Also known as Manhattan distance, city block distance calculates the total length of the path between two points if movement is restricted to horizontal and vertical directions only, much like navigating the street blocks of a city. It sums the absolute differences of their coordinates.
Mathematically, for two points (x₁, y₁) and (x₂, y₂), the City Block distance is:
D_city_block = |x₁ - x₂| + |y₁ - y₂|
- Conceptualization: In the context of digital images or grids, the city block distance metric measures the path between pixels based on a 4-connected neighborhood. This means that only pixels sharing an edge (up, down, left, right) are considered directly adjacent.
- Pixels whose edges touch are 1 unit apart.
- Pixels diagonally touching are 2 units apart using this metric, as it requires two orthogonal steps (one horizontal, one vertical) to reach a diagonal pixel.
Example of City Block Distance
Let's find the City Block distance between point A (1, 1) and point B (3, 3):
- Horizontal difference: |3 - 1| = 2
- Vertical difference: |3 - 1| = 2
- City Block Distance: 2 + 2 = 4
You would need to take 4 steps (e.g., Right, Right, Up, Up) to go from A to B.
Applications of City Block Distance
- Pathfinding algorithms: Often used in simple grid-based games where diagonal movement is not allowed or is penalized.
- Taxicab geometry: A mathematical space where this distance metric is inherent.
- Image processing: Useful for certain types of image analysis, especially when considering connectivity of pixels.
- Layout optimization: In circuit design or facility layout, where components can only be connected orthogonally.
For more details, you can explore the concept of Manhattan Distance on Wikipedia.
Chessboard Distance (Chebyshev Distance)
Commonly referred to as Chebyshev distance, chessboard distance defines the distance between two points as the greatest of their absolute differences along any coordinate dimension. This metric allows for movement along horizontal, vertical, and diagonal paths, similar to how a King moves on a chessboard.
Mathematically, for two points (x₁, y₁) and (x₂, y₂), the Chessboard distance is:
D_chessboard = max(|x₁ - x₂|, |y₁ - y₂|)
- Conceptualization: The chessboard distance metric measures the path between pixels based on an 8-connected neighborhood. This means that any adjacent pixel – whether sharing an edge or a corner (diagonal) – is considered 1 unit away.
Example of Chessboard Distance
Let's find the Chessboard distance between point A (1, 1) and point B (3, 3):
- Horizontal difference: |3 - 1| = 2
- Vertical difference: |3 - 1| = 2
- Chessboard Distance: max(2, 2) = 2
You would need to take 2 steps (e.g., Diagonal-Right-Up, Diagonal-Right-Up) to go from A to B.
Applications of Chessboard Distance
- Image processing: Widely used in digital image analysis, especially for morphological operations and calculating distance transforms where diagonal proximity is important.
- Game development: Useful for character movement in grid-based games where diagonal movement is as "cheap" as orthogonal movement (like a King in chess).
- Robotics: In path planning for robots that can move in any of the 8 directions relative to their current grid cell.
- Maximum absolute error: In numerical analysis, it can represent the maximum error between two vectors.
For further reading, consider the Chebyshev Distance on Wikipedia.
Comparison Table: City Block vs. Chessboard Distance
| Feature | City Block Distance (Manhattan) | Chessboard Distance (Chebyshev) |
|---|---|---|
| Movement Restriction | Orthogonal (horizontal/vertical) only | Any direction (horizontal, vertical, diagonal) |
| Neighborhood | 4-connected neighborhood (edges touch) | 8-connected neighborhood (edges or corners touch) |
| Diagonal Distance | 2 units | 1 unit |
| Formula | |x₁ - x₂| + |y₁ - y₂| |
max(|x₁ - x₂|, |y₁ - y₂|) |
| Common Name | Taxicab Geometry, L₁ norm | King's distance, L_infinity norm |
| Path Appearance | Stair-step, zig-zag | Straight-line, more direct (if diagonals are allowed) |
Practical Insights and Solutions
- Choosing the Right Metric: The choice between City Block and Chessboard distance largely depends on the specific problem's constraints regarding movement and connectivity.
- Use City Block when movement is strictly restricted to orthogonal axes (e.g., finding the shortest path through a maze without diagonal shortcuts, or measuring "turns" in a city grid).
- Use Chessboard when diagonal movements are as permissible and efficient as orthogonal ones (e.g., pathfinding for a chess king, or calculating the maximum deviation in any single dimension).
- Pixel Connectivity: Understanding these distances is crucial for defining pixel connectivity in image processing.
- 4-connectivity is inherent to City Block distance, defining neighbors as those sharing an edge.
- 8-connectivity aligns with Chessboard distance, considering both edge-sharing and corner-sharing pixels as neighbors.
- Performance: Both metrics are computationally inexpensive, involving only absolute differences and sums or maximum operations, making them efficient for large datasets or real-time applications.
