Minimum Moves to Reach Target in Grid

Minimum Moves to Reach Target in Grid

1. What is this problem about?

This problem involves navigating a grid with a unique constraint. You might be given a starting point and a target, but the rules of movement are non-standard. For example, you might be able to move multiple squares at once, or your moves might be limited by a "momentum" or "rotation" mechanic.

In this specific "Hard" variation, the focus is often on mathematical patterns or extremely efficient state-space exploration. The goal is to reach a specific cell (target_x, target_y) from (0, 0) in the minimum number of moves.

2. Why is this asked in interviews?

Bloomberg uses this to test Mathematical Observation and BFS. Key points:

• Standard vs. Non-standard BFS: Is a graph search even necessary, or is there a formula?
• State Definition: What constitutes a unique position?
• Obstacle Management: How to efficiently check for path validity in a large grid.

3. Algorithmic pattern used

The pattern is usually Breadth-First Search (BFS) if there are obstacles, or Math/Greedy if the grid is empty.

• If it's a "Rook-like" or "Queen-like" move: It's often a math problem where you calculate the number of steps to align the x-axis and then the y-axis.
• If there are "forbidden" zones: BFS is the way to go.

4. Example explanation

Start (0,0), Target (2,2), Move: "Can move any number of steps in one cardinal direction."

• Move 1: (0,0) \rightarrow (0,2) (Align y)
• Move 2: (0,2) \rightarrow (2,2) (Align x) Total moves: 2. Even if the target was (100, 100), the answer would still be 2.

5. Common mistakes candidates make

• Over-searching: Using BFS for a problem that can be solved with a simple $O(1)$ calculation.
• Missing Diagonals: If the piece can move diagonally, the answer might be 1.
• Boundary Checks: Forgetting that the grid has limits or that some cells might be blocked.

6. Interview preparation tip

Always ask: "Is this a shortest path in a graph, or is this a geometry problem?" If there are no obstacles, it's usually geometry or math. If there are obstacles, it's almost always BFS.