LeetCode, 𝐃𝐚𝐲-21/365-DSA-Coding 𝐉𝐨𝐮𝐫𝐧𝐞𝐲…Google Prep…

Today I Learned This Problem: Grid Game

Problem Overview

In the Grid Game, two robots compete on a 2-row grid to minimize the maximum score one robot can achieve.

• The first robot moves from left to right, collecting numbers in the top row.
• The second robot starts after the first and moves from left to right, collecting numbers in the bottom row.
  The goal is to calculate the minimum maximum score the second robot can achieve by strategically choosing the point at which the first robot stops.

This problem is an exercise in:

• Dynamic programming or cumulative sums for optimization.
• Iterative comparison of maximum values to find the global minimum.

Challenges Involved

1. Dynamic Score Updates:
   Efficiently tracking the scores for both robots as the first robot moves across the grid.
2. Cumulative Sum Calculation:
   Managing the cumulative sums for both the top and bottom rows to avoid recomputation.
3. Optimization:
   Minimizing the maximum score for the second robot requires careful comparisons and adjustments at each step.

Explanation

Approach to Solve the Problem:

Precompute Cumulative Sums:

• Calculate the total sum of the top row before any moves.
• Start the bottom row sum at 0.

Iterate Across the Grid:

• As the first robot moves from left to right, adjust the remaining score in the top row by subtracting the current value.
• Add the current value to the bottom row cumulative sum.

Compare Scores at Each Step:

• At each point, the second robot’s score will be the maximum of the remaining score in the top row or the accumulated score in the bottom row.
• Update the result to keep track of the minimum maximum score.

Return the Result:

• The minimum of all maximum scores encountered is the answer.

Python Code

class Solution:
    def gridGame(self, grid: List[List[int]]) -> int:
        N = len(grid[0])
        top = sum(grid[0])
        bottom = 0
        result = float('inf')
        
        for i in range(N):
            top -= grid[0][i]
            secondRobot = max(top, bottom)
            result = min(result, secondRobot)
            bottom += grid[1][i]
        
        return result

Other Possible Solutions

Brute Force:

Simulate all possible stopping points for the first robot and calculate the second robot’s score. This approach is inefficient as it involves recomputation of scores at each step.

Dynamic Programming (Alternative):

Using a bottom-up dynamic programming table to calculate scores for both robots. This approach can become complex but is a valid alternative.

Learnings

1. Cumulative Sum Efficiency:
   Precomputing cumulative sums avoids redundant calculations and ensures the solution is performant.
2. Iterative Optimization:
   By dynamically updating and comparing scores at each step, we minimize computational overhead.
3. Real-Life Application:
   Problems like this can model competitive resource allocation or scheduling where minimizing one player’s disadvantage is critical.

Real-Life Applications

1. Resource Allocation:
   Efficiently allocating resources between two competing agents to minimize the disadvantage of one.
2. Game Strategy:
   Optimizing strategies in grid-based competitive games where one player’s score impacts another.
3. Path Optimization:
   Solving problems involving cumulative costs or rewards along a path, similar to logistics or network optimization.

To Sum Up

The Grid Game teaches how to optimize competitive scenarios using dynamic updates and cumulative sums. By focusing on iterative comparisons, the solution minimizes computational effort while ensuring correctness.

Let me know if you have alternative approaches or additional insights!