#2741

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Special Permutations

Count the number of special permutations for a given array using dynamic programming and bit manipulation.

array dynamic programming bit manipulation bitmask

Problem Statement

You are given a 0-indexed array of distinct integers, nums. A permutation of nums is considered special if each element either divides or is divisible by the next element in the permutation. Your task is to return the total number of special permutations of nums, modulo 10^9 + 7.

You are required to find the total number of such special permutations, given that the length of the array is between 2 and 14. The answer can be large, so compute it modulo 10^9 + 7.

Examples

Example 1

Input: nums = [2,3,6]

Output: 2

[3,6,2] and [2,6,3] are the two special permutations of nums.

Example 2

Input: nums = [1,4,3]

Output: 2

[3,1,4] and [4,1,3] are the two special permutations of nums.

Constraints

2 <= nums.length <= 14
1 <= nums[i] <= 109

Solution Approach

Dynamic Programming with Bit Masking

The problem can be solved using dynamic programming with bitmasking. Each bit in the bitmask represents whether an element from the array has been used in the current permutation. Transitions between states are made when an element divides the next one or is divided by it, ensuring that the special permutation condition is met.

State Transitions

The dynamic programming table stores the number of ways to form special permutations for each subset of the array. Starting with an empty set, we progressively add elements, ensuring that the divisibility condition holds between consecutive elements. The state transition is carefully managed using bitmasking to track which elements are used at each step.

Modulo Operation

Since the result can be large, we apply the modulo operation (10^9 + 7) at each step of the dynamic programming solution. This prevents overflow and ensures that the final result fits within the required bounds.

Complexity Analysis

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

The time complexity depends on the approach, but generally, the solution uses dynamic programming and bit masking, making it feasible within the problem constraints. Since the array length is at most 14, the bitmask will have at most 2^14 states, making the approach efficient.

What Interviewers Usually Probe

The candidate demonstrates a clear understanding of dynamic programming with bitmasking.
The candidate successfully applies state transition methods to the problem.
The candidate efficiently manages large numbers using modulo operations.

Common Pitfalls or Variants

Common pitfalls

Forgetting to apply the modulo operation at each step, leading to overflow.
Incorrectly handling the state transitions, such as failing to check the divisibility condition between consecutive elements.
Not efficiently managing bitmasking, resulting in redundant calculations.

Follow-up variants

Considering additional constraints such as larger arrays or different divisibility conditions.
Applying different dynamic programming strategies for state transitions.
Optimizing the solution for faster execution with larger inputs.

FAQ

How can I solve the Special Permutations problem efficiently?

Use dynamic programming with bitmasking to track subsets of the array and apply the divisibility condition during state transitions.

What is the main approach to solving Special Permutations?

The main approach is dynamic programming with bitmasking, where each bit represents an element in the array and the states transition based on divisibility conditions.

What are common mistakes in solving Special Permutations?

Common mistakes include forgetting the modulo operation, mishandling state transitions, and inefficient bitmasking.

How does GhostInterview help in solving the Special Permutations problem?

GhostInterview helps by providing efficient dynamic programming strategies and clarifying the use of bitmasking and modulo operations.

Can this problem be solved without dynamic programming?

While dynamic programming with bitmasking is the most efficient approach, solving the problem without it would likely be computationally expensive and inefficient.

terminal

Solution

Solution 1: State Compression Dynamic Programming

We notice that the maximum length of the array in the problem does not exceed $14$. Therefore, we can use an integer to represent the current state, where the $i$-th bit is $1$ if the $i$-th number in the array has been selected, and $0$ if it has not been selected.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

class Solution:
    def specialPerm(self, nums: List[int]) -> int:
        mod = 10**9 + 7
        n = len(nums)
        m = 1 << n
        f = [[0] * n for _ in range(m)]
        for i in range(1, m):
            for j, x in enumerate(nums):
                if i >> j & 1:
                    ii = i ^ (1 << j)
                    if ii == 0:
                        f[i][j] = 1
                        continue
                    for k, y in enumerate(nums):
                        if x % y == 0 or y % x == 0:
                            f[i][j] = (f[i][j] + f[ii][k]) % mod
        return sum(f[-1]) % mod

Continue Practicing

#2572 count the number of square free subsets #3149 find the minimum cost array permutation #2305 fair distribution of cookies