Minimum Barrier Clearance to Destination

You are navigating a terrain represented as a 0-indexed two-dimensional grid of dimensions m × n. Each cell in this grid contains one of two possible values:

• 0 indicates a passable cell — you can freely move through it without any cost.
• 1 indicates a barrier cell — an obstacle that blocks movement, but can be cleared if necessary.

From any passable cell, you may move in four cardinal directions: up, down, left, or right, as long as the destination cell is within the grid boundaries.

Your objective is to find a path from the top-left corner at position (0, 0) to the bottom-right corner at position (m - 1, n - 1). To accomplish this, you may need to clear some barriers along the way to create a viable path.

Return the minimum number of barriers that must be cleared to establish a traversable route from the starting position to the destination. The starting and ending cells are always guaranteed to be passable (value 0).

Key Observations

• This problem can be modeled as a weighted graph where edges to passable cells have weight 0 and edges to barrier cells have weight 1.
• The goal is to find the path with minimum total weight, which represents the minimum barriers to clear.
• Since edge weights are only 0 or 1, specialized shortest path algorithms like 0-1 BFS using a deque can solve this efficiently.
• Alternatively, Dijkstra's algorithm with a priority queue also works but may have slightly higher overhead for this specific weight structure.

Why This Problem Matters

This is a classic example of weighted shortest path problems on a grid. Understanding how to adapt BFS for weighted graphs (particularly with constrained weights like 0 and 1) is crucial for:

• Navigation systems finding optimal routes
• Game development for pathfinding with obstacles
• Network routing where different links have different costs
• Robotics for motion planning around obstacles