Minimum Knight Moves

Minimum Knight Moves

1. What is this problem about?

The "Minimum Knight Moves" problem is a classic variation of the shortest path problem on a chessboard. Imagine an infinite chessboard where a knight starts at the origin (0, 0). Your goal is to find the minimum number of moves the knight needs to reach a target coordinate (x, y).

A knight moves in an "L" shape: two squares in one cardinal direction (horizontal or vertical) and then one square perpendicularly. This unique movement pattern creates a complex web of potential paths, making it a perfect candidate for exploring graph theory concepts.

2. Why is this asked in interviews?

This question is a staple for companies like Meta, Google, and Microsoft. It's used to evaluate:

• BFS Mastery: Breadth-First Search is the optimal way to find the shortest path in an unweighted grid.
• Symmetry Observation: Can the candidate realize that the number of moves to (x, y) is the same as to (|x|, |y|)? This drastically reduces the search space.
• Optimization Skills: Using techniques like Bidirectional BFS or pruning to handle the potentially infinite grid.

3. Algorithmic pattern used

The primary pattern is Breadth-First Search (BFS).

• State: (curr_x, curr_y).
• Queue: Used to explore all positions at distance 1, then all at distance 2, and so on.
• Visited Set: To avoid redundant calculations and infinite loops.
• Optimization: Since the board is symmetric, we can work only with positive coordinates (abs(x), abs(y)). We can also limit our search to a slightly larger area than the target to allow the knight to "jump over" and back.

4. Example explanation

Target: (2, 1)

1. Start at (0, 0).
2. Possible moves: (2, 1), (1, 2), (-1, 2), (-2, 1), (-2, -1), (-1, -2), (1, -2), (2, -1).
3. One of the moves is directly to (2, 1). Total moves: 1.

Target: (5, 5) From (0, 0), we can't get there in one move. BFS will explore:

• Move 1: (2, 1), (1, 2)...
• Move 2: From (2, 1) to (4, 2), (3, 3)...
• Move 3: From (3, 3) to (5, 4), (4, 5)...
• Move 4: From (5, 4) to (7, 5) or from (4, 5) to (5, 7). Eventually, it will find that (0,0) -> (2,1) -> (4,2) -> (3,4) -> (5,5) (or similar) is a shortest path.

5. Common mistakes candidates make

• Ignoring Symmetry: Searching all four quadrants when searching just one is enough.
• Infinite Board Issues: Not setting a reasonable boundary for the BFS, causing it to wander too far from the target.
• Recursive DFS: Trying to use Depth-First Search to find a "shortest" path, which is highly inefficient and likely to hit a stack overflow.

6. Interview preparation tip

Practice BFS on grids. Understand how to manage a visited set efficiently and how to represent the "L" moves using a simple direction array like dx = [2, 1, -1, -2, -2, -1, 1, 2] and dy = [1, 2, 2, 1, -1, -2, -2, -1].