Minimum Cost Homecoming of a Robot in a Grid

Problem Description

Given an m x n grid, where (0, 0) is the top-left cell and (m - 1, n - 1) is the bottom-right cell, a robot starts at a given position and needs to move to its home position. The robot can move in four directions (left, right, up, down) and incurs costs based on the row and column it moves into. The task is to determine the minimum total cost for the robot to return home.

Key Insights

• The robot can only move vertically or horizontally within the grid.
• Movement costs depend on the row or column that the robot moves into.
• The optimal path can be calculated by determining the total cost of moving from the starting position to the home position along the shortest path.
• The costs for each movement are cumulative and depend on the robot's current row or column.

Space and Time Complexity

Time Complexity: O(|start_row - home_row| + |start_col - home_col|)
Space Complexity: O(1)

Solution

To solve the problem, we can determine the vertical and horizontal distances the robot needs to travel from its starting position to its home position. The total cost can be calculated by summing the costs for each row and column that the robot moves into during its journey.

1. Calculate the vertical distance (row movement) and sum the corresponding row costs for each row traversed.
2. Calculate the horizontal distance (column movement) and sum the corresponding column costs for each column traversed.
3. Return the total cost as the sum of the costs from step 1 and step 2.

Code Solutions