What is the state transition dynamic programming pattern in the Maximum Number of Moves in a Grid problem?

The state transition pattern involves considering each cell as a state, where the number of moves from that state depends on the valid transitions to the right. Dynamic programming is used to propagate the maximum number of moves from each cell to its neighbors.

How does dynamic programming optimize the solution for Maximum Number of Moves in a Grid?

Dynamic programming ensures that each cell's maximum number of moves is calculated only once, significantly reducing redundant calculations and optimizing the solution to run efficiently in O(M * N) time.

What are the main challenges in solving the Maximum Number of Moves in a Grid problem?

The main challenges are managing state transitions efficiently and optimizing space for large grids while maintaining a clear understanding of the movement rules.

How can GhostInterview help with the Maximum Number of Moves in a Grid problem?

GhostInterview provides step-by-step assistance with implementing the state transition dynamic programming approach and helps ensure space optimization for large grids.

What is the time complexity of the Maximum Number of Moves in a Grid solution?

The time complexity of the solution is O(M * N), where M is the number of rows and N is the number of columns in the grid.

#2684

Medium

auto_awesome状态·转移·动态规划

LeetCode 题解工作台

矩阵中移动的最大次数

给你一个下标从 0 开始、大小为 m x n 的矩阵 grid ，矩阵由若干正整数组成。你可以从矩阵第一列中的任一单元格出发，按以下方式遍历 grid ：从单元格 (row, col) 可以移动到 (row - 1, col + 1) 、 (row, col + 1) 和 (row + …

数组动态规划矩阵

题目描述

给你一个下标从 0 开始、大小为 m x n 的矩阵 grid ，矩阵由若干正整数组成。

你可以从矩阵第一列中的任一单元格出发，按以下方式遍历 grid ：

从单元格 (row, col) 可以移动到 (row - 1, col + 1)、(row, col + 1) 和 (row + 1, col + 1) 三个单元格中任一满足值严格大于当前单元格的单元格。

返回你在矩阵中能够移动的最大次数。

示例 1：

输入：grid = [[2,4,3,5],[5,4,9,3],[3,4,2,11],[10,9,13,15]]
输出：3
解释：可以从单元格 (0, 0) 开始并且按下面的路径移动：
- (0, 0) -> (0, 1).
- (0, 1) -> (1, 2).
- (1, 2) -> (2, 3).
可以证明这是能够移动的最大次数。

示例 2：


输入：grid = [[3,2,4],[2,1,9],[1,1,7]]
输出：0
解释：从第一列的任一单元格开始都无法移动。

提示：

m == grid.length
n == grid[i].length
2 <= m, n <= 1000
4 <= m * n <= 10⁵
1 <= grid[i][j] <= 10⁶

lightbulb

解题思路

方法一：BFS

我们定义一个队列 $q$ ，初始时将第一列的所有行坐标加入队列中。

接下来，我们从第一列开始，逐列进行遍历。对于每一列，我们将队列中的所有行坐标依次取出，然后对于每一个行坐标 $i$ ，我们得到其下一列的所有可能行坐标 $k$ ，并且满足 $grid[i][j] < grid[k][j + 1]$ ，将这些行坐标加入到一个新的集合 $t$ 中。如果 $t$ 为空，说明我们无法继续移动，返回当前列数。否则，我们将 $t$ 赋值给 $q$ ，继续下一列的遍历。

最后，如果我们遍历完了所有列，说明我们可以移动到最后一列，返回 $n - 1$ 。

时间复杂度 $O(m \times n)$ ，空间复杂度 $O(m)$ 。其中 $m$ 和 $n$ 分别是矩阵的行数和列数。

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class Solution:
    def maxMoves(self, grid: List[List[int]]) -> int:
        m, n = len(grid), len(grid[0])
        q = set(range(m))
        for j in range(n - 1):
            t = set()
            for i in q:
                for k in range(i - 1, i + 2):
                    if 0 <= k < m and grid[i][j] < grid[k][j + 1]:
                        t.add(k)
            if not t:
                return j
            q = t
        return n - 1

speed

复杂度分析

指标	值
时间	O(M \cdot N)
空间	O(M)

psychology

面试官常问的追问

外企场景

question_mark
Look for understanding of state transition dynamic programming.
question_mark
Evaluate the ability to optimize space by using only two rows in DP.
question_mark
Test the candidate's ability to break down the problem into solvable subproblems and use grid traversal.

warning

常见陷阱

外企场景

error
Not correctly handling the boundary conditions when accessing the grid.
error
Failure to implement an efficient DP solution leading to excessive time complexity.
error
Confusing the state transition rules, such as incorrectly considering moves that violate the grid's conditions.

swap_horiz

进阶变体

外企场景

arrow_right_alt
Consider solving the problem with multiple starting points in the first column.
arrow_right_alt
Optimize for extremely large grids where M or N approach the upper constraint limits.
arrow_right_alt
Solve the problem for grids with a wider variety of values, introducing more complex transition conditions.

help

常见问题

外企场景

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