#3148

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Maximum Difference Score in a Grid

Maximize the score in a grid by making moves to the bottom or right, using state transition dynamic programming.

Problem Statement

You are given an m x n matrix grid consisting of positive integers. Starting from any cell, you can move to another cell that is either directly to the bottom or to the right. The score of each move is calculated as the difference between the value of the destination cell and the value of the current cell.

Your task is to determine the maximum possible score you can achieve by making at least one move. The path is flexible as long as you move down or right, but the goal is to maximize the sum of scores from the moves you make.

Examples

Example 1

Input: grid = [[9,5,7,3],[8,9,6,1],[6,7,14,3],[2,5,3,1]]

Output: 9

We start at the cell (0, 1) , and we perform the following moves: - Move from the cell (0, 1) to (2, 1) with a score of 7 - 5 = 2 . - Move from the cell (2, 1) to (2, 2) with a score of 14 - 7 = 7 . The total score is 2 + 7 = 9 .

Example 2

Input: grid = [[4,3,2],[3,2,1]]

Output: -1

We start at the cell (0, 0) , and we perform one move: (0, 0) to (0, 1) . The score is 3 - 4 = -1 .

Constraints

m == grid.length
n == grid[i].length
2 <= m, n <= 1000
4 <= m * n <= 105
1 <= grid[i][j] <= 105

Solution Approach

State Transition Dynamic Programming

Utilize a dynamic programming approach where each cell stores the maximum score achievable when starting from that cell. For each move, calculate the score based on the difference between the current and destination cells, updating the DP table accordingly.

Grid Traversal

Since moves are restricted to bottom or right directions, start from the bottom-right corner of the grid and work backward. This allows the calculation of the optimal score at each step, avoiding recalculation by utilizing previously computed results.

Path Optimization

To maximize the score, consider the transitions carefully, focusing on moves that give the highest score improvements. Track the best possible score by selecting the most optimal moves from each cell.

Complexity Analysis

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

The time complexity depends on the approach used to calculate the dynamic programming table, typically O(m * n) where m and n are the grid dimensions. Space complexity also typically follows O(m * n) for storing the DP table.

What Interviewers Usually Probe

Tests ability to apply dynamic programming to state transition problems.
Looks for optimization techniques, especially in terms of grid traversal.
Assesses understanding of path selection and maximizing score.

Common Pitfalls or Variants

Common pitfalls

Forgetting to update the DP table in a way that accounts for previously computed optimal scores.
Incorrectly assuming that you can move in any direction, when only bottom or right are allowed.
Not accounting for the minimum move requirement or making unnecessary moves.

Follow-up variants

Different grid sizes or constraints on cell values may require more efficient algorithms.
Variation in starting point could lead to different optimal paths.
Incorporating additional constraints like a limited number of moves could add complexity.

FAQ

What is the state transition dynamic programming approach for this problem?

State transition dynamic programming involves calculating the score of each move based on previously computed results, updating the table to reflect the maximum score achievable from each cell.

Can I move in directions other than right or down in the grid?

No, you are only allowed to move to the bottom or right cells in the grid.

How does the maximum score calculation depend on previous moves?

The score of each move is calculated based on the difference between the current and destination cell values, with previous moves impacting the score accumulation.

What’s the best strategy to maximize the score in the grid?

The best strategy involves selecting moves that give the highest possible score difference while ensuring that moves are restricted to the allowed directions (right or down).

What are common mistakes people make in this problem?

A common mistake is not properly updating the dynamic programming table to track the optimal path, or misinterpreting the allowed directions for movement.

terminal

Solution

Solution 1: Dynamic Programming

According to the problem description, if the values of the cells we pass through are $c_1, c_2, \cdots, c_k$, then our score is $c_2 - c_1 + c_3 - c_2 + \cdots + c_k - c_{k-1} = c_k - c_1$. Therefore, the problem is transformed into: for each cell $(i, j)$ of the matrix, if we take it as the endpoint, what is the minimum value of the starting point.

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class Solution:
    def maxScore(self, grid: List[List[int]]) -> int:
        f = [[0] * len(grid[0]) for _ in range(len(grid))]
        ans = -inf
        for i, row in enumerate(grid):
            for j, x in enumerate(row):
                mi = inf
                if i:
                    mi = min(mi, f[i - 1][j])
                if j:
                    mi = min(mi, f[i][j - 1])
                ans = max(ans, x - mi)
                f[i][j] = min(x, mi)
        return ans

Continue Practicing

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