Minimum Cost Path with Alternating Directions I - Solution & Explanation

Problem Overview: You need the minimum cost path from the start to the destination while strictly alternating movement directions. Consecutive moves cannot use the same direction, so the path must follow a fixed alternating pattern such as right–down–right–down or down–right–down–right.

Approach 1: Brute Force Path Enumeration (Exponential Time, O(2^n) time, O(n) space)

A direct way is to recursively generate every valid path while enforcing the alternating direction rule. At each step, you attempt a move only if it differs from the previous direction. Track the accumulated cost and update the minimum once the destination is reached. This guarantees correctness but becomes infeasible quickly because the number of paths grows exponentially with the grid size. This approach mainly helps validate edge cases and understand how the alternating constraint restricts valid paths.

Approach 2: Brain Teaser Pattern Evaluation (O(n) time, O(1) space)

The alternating rule drastically limits the number of possible paths. Instead of exploring many routes, observe that only two valid direction sequences can exist: starting with the first direction or starting with the second direction. For example, a path might follow R → D → R → D or D → R → D → R. Once the starting direction is chosen, the entire sequence becomes fixed because every step must alternate.

Iterate along the path defined by each possible pattern and accumulate the total cost of visited cells. If a pattern cannot reach the destination due to mismatched required moves, discard it. The minimum cost among the valid patterns is the answer. This turns the problem into a simple simulation rather than a search.

This technique works because the alternating constraint removes the combinatorial explosion typically seen in grid path problems. Instead of dynamic programming over many states, you evaluate at most two deterministic paths. The reasoning is closer to a brainteaser than a traditional graph traversal. The arithmetic and step counting aspects also connect to common math interview puzzles where constraints reduce possibilities dramatically.

Recommended for interviews: The brain teaser approach is what interviewers expect. Showing the brute-force idea first demonstrates that you understand the search space, but recognizing that only two alternating patterns exist shows strong problem‑solving intuition and reduces the complexity to linear time.