Minimum Knight Moves

You have an infinite chessboard. A Knight starts at coordinates (0, 0).

• Goal: Return the minimum number of steps to move the Knight to square (x, y).
• Moves: The standard 'L' shape: (±1, ±2) or (±2, ±1).

This is a Shortest Path on an Unweighted Graph problem.

• Nodes: Coordinates (row, col).
• Edges: The 8 possible Knight jumps.

Since edges are unweighted (cost = 1 move), BFS is the optimal choice.

Optimizations:

1. Symmetry: Moving to (x, y) takes the same steps as (-x, -y), (x, -y), etc. We can use abs(x), abs(y) and restrict our search primarily to the first quadrant (+2 buffer).
2. Bi-directional BFS: (Advanced) Expand from Start and Target efficiently to meet in the middle.

Watch how the Knight's possible moves expand layer-by-layer like a ripple.