#2318

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Number of Distinct Roll Sequences

Calculate the number of distinct sequences of dice rolls based on specific conditions using dynamic programming.

Problem Statement

Given an integer n, you roll a fair 6-sided die n times. You need to calculate the total number of distinct sequences of rolls, where two sequences are considered distinct if at least one roll differs.

To ensure the validity of the sequence, two specific conditions must hold: first, for each pair of rolls at distinct positions i and j (1-indexed), the absolute difference between the values of the rolls at i and j must be greater than 1; second, the greatest common divisor of the rolls at positions i and j must be 1. Return the total number of valid sequences modulo 10^9 + 7.

Examples

Example 1

Input: n = 4

Output: 184

Some of the possible sequences are (1, 2, 3, 4), (6, 1, 2, 3), (1, 2, 3, 1), etc. Some invalid sequences are (1, 2, 1, 3), (1, 2, 3, 6). (1, 2, 1, 3) is invalid since the first and third roll have an equal value and abs(1 - 3) = 2 (i and j are 1-indexed). (1, 2, 3, 6) is invalid since the greatest common divisor of 3 and 6 = 3. There are a total of 184 distinct sequences possible, so we return 184.

Example 2

Input: n = 2

Output: 22

Some of the possible sequences are (1, 2), (2, 1), (3, 2). Some invalid sequences are (3, 6), (2, 4) since the greatest common divisor is not equal to 1. There are a total of 22 distinct sequences possible, so we return 22.

Constraints

1 <= n <= 104

Solution Approach

Dynamic Programming

Use a dynamic programming approach to keep track of valid sequences. Maintain an array that stores the number of valid sequences for each possible die roll at each position. Transition through each state by checking the conditions and updating the array based on previous values.

State Transition

Each roll introduces a state based on the number rolled. Transition between states by considering the constraints on differences and GCD values, ensuring only valid sequences are counted. State transitions must avoid repeating dice values and must respect the GCD condition.

Modulo Operation

Since the result may be large, use modulo 10^9 + 7 throughout the calculations to prevent overflow and ensure the final result fits within the limits of typical integer sizes.

Complexity Analysis

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

The time complexity depends on the dynamic programming approach chosen, where each step involves checking previous states and updating based on possible roll values. The space complexity depends on the storage needed for the DP array, which is related to n and the number of states for each roll.

What Interviewers Usually Probe

Look for an understanding of dynamic programming and state transitions.
Check if the candidate correctly identifies the GCD condition and its impact on sequence validity.
Evaluate if the candidate is able to optimize the solution using modulo 10^9 + 7.

Common Pitfalls or Variants

Common pitfalls

Forgetting to handle the GCD condition correctly, resulting in incorrect valid sequences.
Failing to use dynamic programming efficiently, leading to excessive time complexity.
Not applying modulo 10^9 + 7 consistently, leading to integer overflow or incorrect answers.

Follow-up variants

Use memoization to optimize the dynamic programming solution.
Extend the problem to consider rolling a die with more than six sides.
Consider optimizing space by reducing the state storage for previous rolls.

FAQ

What is the primary approach to solve the Number of Distinct Roll Sequences problem?

The primary approach is dynamic programming with state transitions, considering both GCD constraints and roll differences.

How do I handle large numbers in the Number of Distinct Roll Sequences problem?

Use modulo 10^9 + 7 to ensure the results do not exceed the integer limits and avoid overflow.

Can I optimize the dynamic programming solution?

Yes, memoization can be used to optimize repeated calculations, and space complexity can be reduced by storing only the relevant states.

What is the time complexity of the Number of Distinct Roll Sequences problem?

The time complexity depends on the specific dynamic programming approach, where each step involves checking and transitioning between possible states.

What are the common mistakes when solving the Number of Distinct Roll Sequences problem?

Common mistakes include incorrect handling of the GCD condition, inefficient dynamic programming implementations, and neglecting to apply the modulo operation.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def distinctSequences(self, n: int) -> int:
        if n == 1:
            return 6
        mod = 10**9 + 7
        dp = [[[0] * 6 for _ in range(6)] for _ in range(n + 1)]
        for i in range(6):
            for j in range(6):
                if gcd(i + 1, j + 1) == 1 and i != j:
                    dp[2][i][j] = 1
        for k in range(3, n + 1):
            for i in range(6):
                for j in range(6):
                    if gcd(i + 1, j + 1) == 1 and i != j:
                        for h in range(6):
                            if gcd(h + 1, i + 1) == 1 and h != i and h != j:
                                dp[k][i][j] += dp[k - 1][h][i]
        ans = 0
        for i in range(6):
            for j in range(6):
                ans += dp[-1][i][j]
        return ans % mod

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