#1411

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Number of Ways to Paint N × 3 Grid

Calculate the number of ways to paint a grid of size n x 3 with distinct adjacent colors using dynamic programming.

Problem Statement

You are given a grid of size n x 3 and need to paint each cell using one of three colors: Red, Yellow, or Green. The goal is to ensure that no two adjacent cells (either horizontally or vertically) have the same color.

Given n, the number of rows in the grid, return the number of ways you can paint the grid while respecting the adjacency condition. As the answer may grow large, compute it modulo 10^9 + 7.

Examples

Example 1

Input: n = 1

Output: 12

There are 12 possible way to paint the grid as shown.

Example 2

Input: n = 5000

Output: 30228214

Example details omitted.

Constraints

n == grid.length
1 <= n <= 5000

Solution Approach

State Transition Dynamic Programming

The solution relies on dynamic programming where the state of each row depends on the previous row's configuration. We maintain states for the previous row's valid colorings and use transitions to calculate the next row's possibilities.

Recursive Formula for Transitions

For each cell in the row, we calculate possible valid configurations by transitioning from the previous row. The recursive relations help derive the number of valid ways to paint the grid by ensuring that no two adjacent cells have the same color.

Modulo Operation

Due to the large number of possible configurations, apply the modulo operation (10^9 + 7) to the result at every step to prevent integer overflow and meet the problem’s constraints.

Complexity Analysis

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

The time complexity depends on the approach used but generally, it will be O(n) since each row can be computed in constant time after calculating the previous row. Space complexity also depends on the approach, typically O(1) or O(n) for storing intermediate results, depending on whether we maintain only the last row or all rows.

What Interviewers Usually Probe

Ability to identify dynamic programming state transitions.
Understanding of how to apply modulo operations in large-number problems.
Proficiency in applying recursive relations to problems involving grid configurations.

Common Pitfalls or Variants

Common pitfalls

Failing to consider the adjacency constraint, leading to invalid configurations.
Not using the modulo operation correctly, potentially resulting in overflow errors.
Misunderstanding dynamic programming transitions, causing incorrect state updates.

Follow-up variants

Increase grid size to n x m, requiring generalized state transition management.
Implement an approach that handles multiple constraints, such as limiting the number of color changes in each row.
Consider variations with additional constraints like coloring each row in a specific pattern or requiring fewer colors.

FAQ

What is the key approach to solving the Number of Ways to Paint N × 3 Grid problem?

The key approach is state transition dynamic programming, where the solution for each row depends on the configuration of the previous row, ensuring adjacent cells have different colors.

How do we avoid integer overflow in the Number of Ways to Paint N × 3 Grid problem?

You should apply the modulo operation (10^9 + 7) at every step of the calculation to avoid overflow and keep the result within the problem's constraints.

What does the dynamic programming state represent in the Number of Ways to Paint N × 3 Grid problem?

The dynamic programming state represents the number of valid colorings of a given row, considering the color constraints from the previous row.

How does GhostInterview help in solving dynamic programming problems like Number of Ways to Paint N × 3 Grid?

GhostInterview offers step-by-step guidance through the dynamic programming approach, identifying common mistakes and helping to optimize the solution.

What are the common pitfalls in solving the Number of Ways to Paint N × 3 Grid problem?

Common pitfalls include ignoring adjacency constraints, not applying the modulo operation correctly, and incorrectly managing state transitions in the dynamic programming approach.

terminal

Solution

Solution 1: Recursion

We classify all possible states for each row. According to the principle of symmetry, when a row only has $3$ elements, all legal states are classified as: $010$ type, $012$ type.

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class Solution:
    def numOfWays(self, n: int) -> int:
        mod = 10**9 + 7
        f0 = f1 = 6
        for _ in range(n - 1):
            g0 = (3 * f0 + 2 * f1) % mod
            g1 = (2 * f0 + 2 * f1) % mod
            f0, f1 = g0, g1
        return (f0 + f1) % mod

Solution 2: State Compression + Dynamic Programming

We notice that the grid only has $3$ columns, so there are at most $3^3=27$ different coloring schemes in a row.

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class Solution:
    def numOfWays(self, n: int) -> int:
        mod = 10**9 + 7
        f0 = f1 = 6
        for _ in range(n - 1):
            g0 = (3 * f0 + 2 * f1) % mod
            g1 = (2 * f0 + 2 * f1) % mod
            f0, f1 = g0, g1
        return (f0 + f1) % mod

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