Lattice Path
- A lattice path moves along grid lines, taking only horizontal or vertical steps to adjacent cells.
- Common in combinatorial problems for counting the number of ways to reach a point or for optimization tasks like finding the shortest path.
- Can be generalized to three-dimensional grids (adding up and down moves along the z-axis).
Definition
Section titled “Definition”A lattice path is a path that moves only horizontally or vertically on a grid. The path follows the grid lines, meaning it can only move to adjacent cells that share an edge.
Explanation
Section titled “Explanation”Lattice paths are used in combinatorial problems where the objective is to count the number of ways to reach a specific point on the grid. Movement is constrained to the grid lines (horizontal and vertical directions), and combinatorial techniques are applied to determine counts of possible paths. The concept extends to three-dimensional grids (for example, a cube), where moves along the z-axis (up and down) are also allowed and counted using similar combinatorial methods. Lattice path formulations are also useful for optimization problems such as finding the shortest path between two points on the grid.
Examples
Section titled “Examples”Knight on a chessboard
Section titled “Knight on a chessboard”One example of a lattice path is the movement of a knight on a chessboard. The knight can move in an “L” shape, meaning it can move two squares horizontally and one square vertically, or two squares vertically and one square horizontally. This movement pattern is constrained to the grid of the chessboard, making it a lattice path. The number of possible moves the knight can make from a certain position can be determined using combinatorial techniques.
Robot on a grid of streets
Section titled “Robot on a grid of streets”Another example of a lattice path is the movement of a robot on a grid of streets. The robot can only move north, south, east, or west on the grid, following the lines of the streets. This movement is also constrained to the grid, making it a lattice path. The number of possible paths the robot can take from one point on the grid to another can be determined using combinatorial techniques.
Three-dimensional grids
Section titled “Three-dimensional grids”Lattice paths can also be extended to three-dimensional grids, such as a cube. In this case, the path can move in the horizontal and vertical directions, as well as up and down along the z-axis. The number of possible paths on a three-dimensional grid can also be determined using combinatorial techniques.
Use cases
Section titled “Use cases”- Counting the number of possible paths between points on a grid (combinatorial enumeration).
- Optimization problems on grids, such as finding the shortest path between two points.
Related terms
Section titled “Related terms”- Combinatorial problems / Combinatorics
- Shortest path
- Grid