2741. Special Permutations

Problem Description

In this problem, we are given a zero-indexed array nums that consists of n distinct positive integers. We are asked to find the number of permutations of nums that are "special." A permutation is considered special if it meets the following condition: for every adjacent pair of elements in the permutation (nums[i], nums[i+1]) where 0 <= i < n - 1, the pair must satisfy either nums[i] % nums[i+1] == 0 or nums[i+1] % nums[i] == 0. In other words, each adjacent pair must be such that one number is a divisor of the other. We need to return the total number of such special permutations modulo 10^9 + 7 to handle the large number that might result from the calculation.

Intuition

To solve this problem, we can use dynamic programming with bitmasking to handle the permutations efficiently. We consider each bit in a bitmask to represent whether a particular element of the array nums has already been used in the permutation or not. Since nums contains n distinct elements, we can represent the state of the permutation using n bits, which leads to a total of 2^n states (m in the code).

The main idea behind the solution is the following steps:

1. Initialize a 2D array f with dimensions m * n to store the number of ways to achieve a valid permutation using some of the elements of nums. The row represents the bitmask (which elements have been used), and the column j represents the last element used in the permutation.

2. Loop through all possible bitmask states i, and for each, we calculate the number of valid special permutations ending with each element x (where i has the jth bit set, indicating that x is the last element of the permutation). To do this:

   • Find ii by subtracting the current element x from the bitmask, resulting in the bitmask of the rest of the elements.

   • If ii equals zero, it means that this is the starting element of the permutation; set f[i][j] to 1.

   • Otherwise, iterate over the array again using index k and element y representing the second-to-last element in the permutation, if the current element x can divide y or vice versa, increment the count in f[i][j] by the number of ways you can form a permutation that ends with y; update it as f[i][j] = (f[i][j] + f[ii][k]) % mod.

3. After filling up the table, the sum of counts in the last row of f will give the total number of special permutations, because the last row represents the state where all elements have been used.

This dynamic programming approach effectively counts all valid permutations while ensuring that we do not revisit configurations we have already considered, which greatly optimizes the solution.

Learn more about Bitmask patterns.

Solution Approach

The solution provided uses dynamic programming and bitmasking to find the number of special permutations.

Here's the breakdown of the implementation:

• First, the mod variable is set to 10^9 + 7 to ensure all calculations are done under modulo arithmetic to prevent integer overflow.

• The variable n is assigned the length of the nums array. And m is calculated as 1 << n, which equals 2^n and represents the total number of states each element can be in (used or unused).

• The f array is initialized as a two-dimensional list of size m * n, with all elements set to zero. This array holds the number of special permutations possible for each combination of elements.

• The outermost loop iterates over all possible states, represented by the bitmask i. The bitmask i has n bits, with each bit corresponding to whether a particular number in nums has been included in the permutation (1) or not (0).

• Within the outer loop, we iterate over all elements x in nums with their corresponding index j. If the jth bit of bitmask i is set (i >> j & 1), it indicates that number x is being considered as the last number in the permutation.

• For each such x, we calculate ii, which is the bitmask state without x (done by i ^ (1 << j)).

• If ii is zero (ii == 0), it means we are considering permutations where x is the only number so far. Thus, there is only one way to form such a permutation and f[i][j] is set to 1.

• If ii is not zero, we look at all other numbers y in nums that could come before x in the permutation. We use the second-to-last number's k index to check divisibility between y and x (either x % y == 0 or y % x == 0). If the divisibility condition is met, it means y can come before x in a permutation. We add the count of special permutations ending with y (stored in f[ii][k]) to the count of permutations ending with x (f[i][j]), applying modulo every time to keep numbers within bounds.

• After completing both loops, the array f[-1] contains counts of all special permutations for each nums element as the last number. The sum of all these counts gives us the total number of special permutations. The final result is taken modulo 10^9 + 7 to get the answer within the required bounds.

This algorithm is efficient because it takes advantage of the structure of permutations and divisibility to prune the search space and count only valid special permutations, avoiding the need to generate and test every permutation explicitly, which would be impractical for large n.

Example Walkthrough

Consider an array nums with distinct positive integers like [1, 2, 3]. Let's walk through the solution approach using this set of numbers to understand how the dynamic programming and bitmasking technique is applied.

1. First, we set n equal to the length of nums, which in this case is 3. We calculate m as 1 << n (which is 2^n), amounting to 8 for this example because there are 2^3 possible states for three distinct numbers.

2. Initialize the DP array f with size m * n. In this case, f will be an 8 by 3 array filled initially with zeros. The state of f will look as follows after initialization, where f[i][j] represents the number of ways to form permutations of size i with j as the last number:

   1f = [
   2  [0, 0, 0],  // state 0: no elements used
   3  [0, 0, 0],  // state 1: only num[0] used
   4  [0, 0, 0],  // state 2: only num[1] used
   5  [0, 0, 0],  // state 3: num[0] and num[1] used
   6  [0, 0, 0],  // state 4: only num[2] used
   7  [0, 0, 0],  // state 5: num[0] and num[2] used
   8  [0, 0, 0],  // state 6: num[1] and num[2] used
   9  [0, 0, 0],  // state 7: all numbers used
   10]

3. Start iterating over all possible states (i) from 1 to m - 1. For each state i, we look for all numbers x in nums that could potentially be the last number in the permutation.

4. For each number x (with index j), if x is included in state i (checked by i >> j & 1), we calculate the state ii by removing x from i (via i ^ (1 << j)).

5. If ii is 0, we know that x is at the start of the permutation, hence f[i][j] is set to 1.

6. If ii is not 0, check all numbers y (with index k) that are not x and occur before x in the permutation. If x divides y or y divides x, add the number of permutations that end with y (f[ii][k]) to the current permutation count (f[i][j]).

Applying this process to our example, let's consider the state where i = 3 (binary 011), which means nums[0] (value 1) and nums[1] (value 2) are included. Since 1 is a divisor of 2, we can form a permutation [1, 2]. Thus, f[3][1] (where 3 represents the state 011 and 1 represents index of number 2 in nums) will increment by 1. Iterate this procedure for each state and number pair.

Once the table is filled, sum up the last row's values to get the total number of special permutations. For our example, f[7] (which represents all numbers being used) yields the count for each number as the last in a special permutation. Since all numbers in nums are divisible by 1, any permutation where 1 is the first element will be valid, leading to the total count being the sum of the last row of f modulo 10^9 + 7.

This small example demonstrates the use of dynamic programming and bitmasking for efficiently calculating the number of special permutations. Each state is systematically built upon the previous states, ensuring that all permutations are accounted for and that each permutation is valid according to the given divisibility condition.

Solution Implementation

Time and Space Complexity

The given Python function specialPerm calculates a special permutation count under certain constraints. We'll analyze the time and space complexity of this function.

Time Complexity

The function involves multiple nested loops:

• The outermost loop runs through all subsets of indices formed from the nums array, having a total of 2^n iterations (n is the length of nums), because m = 1 << n which is 2^n.
• The second loop iterates over each element in nums, contributing a factor of n.
• The innermost loop runs conditionally when the bit at position j is set in the current subset i (i >> j & 1). When this condition is met, it attempts to iterate over nums again to update f[i][j]. However, this does not bring another factor of n because it will only iterate up to n times in the worst-case scenario, not for each subset or combination.

Considering the bit manipulation operations and modulo operations inside the loops are constant time, the time complexity is derived primarily from these loops.

Therefore, the total time complexity is O(n * 2^n) since the two significant factors are the iterations over the subsets (2^n) and the elements in nums (n).

Space Complexity

The space usage in the function comes from:

• The list f: a 2D list of dimensions 2^n by n. It holds the count of permutations that end with each element j for all subsets of nums.
• Auxiliary space: negligible constant space used for loop indices and temporary variables.

The space complexity is thus determined by the size of f, which is O(n * 2^n).

In conclusion, the function has a time complexity of O(n * 2^n) and a space complexity of O(n * 2^n).

Learn more about how to find time and space complexity quickly using problem constraints.