#2684

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Maximum Number of Moves in a Grid

Maximize the number of moves in a grid starting from the first column using state transition dynamic programming.

Problem Statement

Given an m x n grid of positive integers, you can start at any cell in the first column. From there, you must traverse the grid according to the following rules: at each step, you may move to an adjacent cell in the row to the right if its value is strictly greater than the current cell's value. The goal is to determine the maximum number of valid moves you can make starting from any cell in the first column.

For example, in a grid like [[2,4,3,5],[5,4,9,3],[3,4,2,11],[10,9,13,15]], starting from cell (0,0), you can move right to (0,1), then to (1,2), and finally to (2,3). This results in 3 valid moves. The task is to compute the maximum such moves in the grid.

Examples

Example 1

Input: grid = [[2,4,3,5],[5,4,9,3],[3,4,2,11],[10,9,13,15]]

Output: 3

We can start at the cell (0, 0) and make the following moves:

(0, 0) -> (0, 1).
(0, 1) -> (1, 2).
(1, 2) -> (2, 3). It can be shown that it is the maximum number of moves that can be made.

Example 2

Input: grid = [[3,2,4],[2,1,9],[1,1,7]]

Output: 0

Starting from any cell in the first column we cannot perform any moves.

Constraints

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

Solution Approach

Dynamic Programming Approach

The optimal solution uses dynamic programming (DP) where each cell stores the maximum number of moves starting from that position. Starting from the last column, propagate values towards the first column, calculating the maximum moves by considering valid moves to the right. The DP approach ensures we efficiently compute results without redundant recalculations.

State Transition Representation

The key to solving this problem is recognizing the state transition pattern. The value of a cell represents the number of moves that can be made from it, and transitions occur only if the current cell's value is smaller than the adjacent right cell's value. By applying DP to handle state transitions efficiently, we can track the best path for each cell.

Time and Space Optimization

The time complexity of the solution is O(M * N), where M and N are the grid dimensions. To optimize space, store only the DP values of the current and previous rows, reducing space complexity to O(M), which is crucial for larger grids within the problem's constraints.

Complexity Analysis

Metric	Value
Time	O(M \cdot N)
Space	O(M)

The algorithm computes the DP values for each cell, visiting every cell exactly once, making the time complexity O(M * N), where M is the number of rows and N is the number of columns. Space complexity is reduced to O(M) by using a row-based DP table, maintaining only the current and previous row values at any given time.

What Interviewers Usually Probe

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

Common Pitfalls or Variants

Common pitfalls

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

Follow-up variants

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

FAQ

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.

terminal

Solution

Solution 1: BFS

We define a queue $q$, and initially add all the row coordinates of the first column to the queue.

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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

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