Minimum Cost Path with Alternating Directions II

The problem involves finding the minimum cost path through a grid where each cell has a cost. You can move either right or down and alternate directions. Dynamic programming (DP) is a key technique to solve it optimally, focusing on state transitions between grid cells.

Problem Statement

You are given a grid with dimensions m x n, where m and n represent the number of rows and columns, respectively. Each cell (i, j) has a cost defined by the formula (i + 1) * (j + 1), and you are provided a 2D array waitCost where waitCost[i][j] gives the cost to wait on that cell. Your goal is to find the minimum cost to traverse the grid from the top-left corner to the bottom-right corner, alternating between right and down moves at each step.

You are allowed to move either to the right or down, and you must alternate between the two directions at each step. You must minimize the total cost of traveling from the top-left corner to the bottom-right corner, considering both the cost of the cell you are currently on and the waitCost for waiting in each cell.

Examples

Example 1

Input: m = 1, n = 2, waitCost = [[1,2]]

Output: 3

The optimal path is: Thus, the total cost is 1 + 2 = 3 .

Example 2

Input: m = 2, n = 2, waitCost = [[3,5],[2,4]]

Output: 9

The optimal path is: Thus, the total cost is 1 + 2 + 2 + 4 = 9 .

Example 3

Input: m = 2, n = 3, waitCost = [[6,1,4],[3,2,5]]

Output: 16

The optimal path is: Thus, the total cost is 1 + 2 + 1 + 4 + 2 + 6 = 16 .

Constraints

• 1 <= m, n <= 105
• 2 <= m * n <= 105
• waitCost.length == m
• waitCost[0].length == n
• 0 <= waitCost[i][j] <= 105

Solution Approach

State Transition Dynamic Programming

Use dynamic programming to track the minimum cost to reach each cell in the grid. Maintain a DP array where each entry represents the minimal cost to reach that cell, considering both the base cost and the alternating direction costs. For each move, transition to the next cell in the optimal direction (either right or down), updating the DP array accordingly.

Alternating Movement

To handle the alternating movement constraint, ensure that each DP transition considers the direction of the previous move. When transitioning from one cell to the next, alternate between moving right and down while updating the DP array to reflect the accumulated minimal cost at each step.

Boundary Handling

Properly handle the grid boundaries when moving right or down, ensuring that no invalid moves are made outside the grid. Ensure that the DP transitions respect the grid limits by checking bounds at each step to avoid index errors.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity of this solution is O(m * n) because each cell in the grid is processed once. The space complexity is O(m * n) due to the DP array used to store the minimum cost for each cell. Optimizations may reduce space usage, but the time complexity remains dependent on the grid size.

What Interviewers Usually Probe

• Can the candidate effectively apply dynamic programming to solve grid-based problems?
• Does the candidate handle alternating movement correctly and efficiently?
• How well does the candidate handle edge cases, like grid boundaries and minimal grid sizes?

Common Pitfalls or Variants

Common pitfalls

• Failing to alternate directions correctly can lead to suboptimal solutions.
• Not accounting for grid boundaries and trying to access out-of-bounds cells.
• Overcomplicating the solution by missing the optimal DP approach and using unnecessary nested loops or recursion.

Follow-up variants

• What if there are obstacles or restricted cells that cannot be passed?
• How would the solution change if movement was allowed diagonally as well?
• How can this problem be adapted to use a priority queue or greedy approach for pathfinding?

FAQ

What is the primary pattern used to solve the Minimum Cost Path with Alternating Directions II problem?

The primary pattern for this problem is state transition dynamic programming, where each state represents the minimum cost to reach a cell while alternating directions.

How do I minimize the cost in a grid while alternating directions?

You minimize the cost by using dynamic programming to track the minimal cost to reach each cell, considering both the base cost and the waitCost for each cell.

What are the constraints for the Minimum Cost Path with Alternating Directions II problem?

The constraints are 1 <= m, n <= 105, 2 <= m * n <= 105, waitCost.length == m, waitCost[0].length == n, and 0 <= waitCost[i][j] <= 105.

What happens if I don't alternate directions correctly?

Failing to alternate directions correctly will lead to an incorrect path and a suboptimal solution, which will not minimize the cost as intended.

Can this problem be solved with recursion instead of dynamic programming?

While recursion can be used, dynamic programming is more efficient because it avoids redundant calculations by storing intermediate results.