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Prime Arrangements

Easy

Asked by:

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Topics:

ArraysDynamic ProgrammingGreedy Algorithms

Return the number of permutations of 1 to n so that prime numbers are at prime indices (1-indexed.)

(Recall that an integer is prime if and only if it is greater than 1, and cannot be written as a product of two positive integers both smaller than it.)

Since the answer may be large, return the answer modulo 10^9 + 7.

Example 1:

Input: n = 5
Output: 12
Explanation: For example [1,2,5,4,3] is a valid permutation, but [5,2,3,4,1] is not because the prime number 5 is at index 1.

Example 2:

Input: n = 100
Output: 682289015

Constraints:

1 <= n <= 100

Try coding it in LeetCode

Prime Arrangements

Easy

Asked by:

35 views

Topics:

ArraysDynamic ProgrammingGreedy Algorithms

Return the number of permutations of 1 to n so that prime numbers are at prime indices (1-indexed.)

(Recall that an integer is prime if and only if it is greater than 1, and cannot be written as a product of two positive integers both smaller than it.)

Since the answer may be large, return the answer modulo 10^9 + 7.

Example 1:

Input: n = 5
Output: 12
Explanation: For example [1,2,5,4,3] is a valid permutation, but [5,2,3,4,1] is not because the prime number 5 is at index 1.

Example 2:

Input: n = 100
Output: 682289015

Constraints:

1 <= n <= 100

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the maximum possible value of `n`? I need to understand the scale for prime number calculation.
Is `n` always a positive integer?
Should the result be returned modulo a specific number? If so, what is that number?
Could you define more precisely what is meant by a 'valid arrangement'? Is it sufficient for the primes to be in prime positions, and non-primes to be in non-prime positions, or are there additional constraints?
Is there any specified behavior expected for `n` equal to 0 or 1?

Brute Force Solution

Approach

The brute force approach to Prime Arrangements involves trying all possible orderings of numbers to see which ones satisfy the prime/non-prime placement rules. We generate every permutation and then check if that permutation is valid. It's like trying every single possibility, one after the other, until we find the right answer.

Here's how the algorithm would work step-by-step:

First, figure out all the different ways you can arrange the numbers from 1 to the input number.
For each of these arrangements, go through each position and check if a prime number is in a prime position and a non-prime number is in a non-prime position.
If, for a particular arrangement, a prime number is in a non-prime position or vice versa, then that arrangement is incorrect, and you should disregard it.
If an arrangement follows the rule where primes are in prime positions and non-primes are in non-prime positions, then count that as a valid arrangement.
After checking all possible arrangements, report the total number of valid arrangements.

Code Implementation

def prime_arrangements_brute_force(number):
    all_permutations = list(permutations(range(1, number + 1)))
    valid_arrangements_count = 0
    
    for permutation in all_permutations:
        is_valid = True
        
        for index, number_in_permutation in enumerate(permutation):
            # Check if prime is at a non-prime position or vice versa
            if is_prime(index + 1) != is_prime(number_in_permutation):

                is_valid = False
                break
        
        # Only increment count if permutation is valid
        if is_valid:
            valid_arrangements_count += 1
            
    return valid_arrangements_count % (10**9 + 7)

from itertools import permutations

def is_prime(number_to_check):
    if number_to_check <= 1:
        return False
    for i in range(2, int(number_to_check**0.5) + 1):
        if number_to_check % i == 0:
            return False
    return True

Big(O) Analysis

Time Complexity

O(n!) – Generating all permutations of n numbers is the dominant factor. There are n! (n factorial) possible permutations. For each permutation, we iterate through the n positions to check if the primes are in the prime positions. The primality test can be considered constant time because we pre-calculate primes up to n. Thus the time complexity is dominated by generating and validating the permutations, leading to a total time complexity of O(n!).

Space Complexity

O(N!) – The brute force approach generates all possible permutations of numbers from 1 to N. Storing each permutation requires an array of size N. Since there are N! permutations, the algorithm implicitly stores up to N! different arrays of size N simultaneously. This leads to a space complexity of O(N * N!). However, if we are only concerned about the space used at any *given* time during the permutation generation process (assuming the permutations are processed one at a time without being saved), the dominant space usage would come from the recursion stack for generating these permutations, which requires O(N) space. But based on the plain English which speaks to gathering all possible permutations and checking them, it implies storing multiple arrangements. Therefore, O(N!) is more accurate.

Optimal Solution

Approach

The goal is to arrange numbers from 1 to a given number 'n' such that prime numbers are at prime positions and non-prime numbers are at non-prime positions. The core idea is to count the number of primes and non-primes separately, then calculate the factorial of each group, and multiply them together, handling potential large numbers using modulo.

Here's how the algorithm would work step-by-step:

First, count how many prime numbers there are between 1 and 'n'.
Then, figure out how many non-prime numbers there are; this is just 'n' minus the number of primes.
Calculate the factorial of the number of prime numbers. This tells you how many ways you can arrange the primes in the prime positions.
Calculate the factorial of the number of non-prime numbers. This tells you how many ways you can arrange the non-primes in the non-prime positions.
Multiply these two factorials together. This gives you the total number of valid arrangements.
Since the numbers can get very big, perform the multiplication and factorial calculations using modulo to avoid exceeding the maximum number size. This keeps the result within a manageable range.

Code Implementation

def prime_arrangement(n): 
    modulo = 10**9 + 7

    def count_primes(n):
        if n < 2:
            return 0
        is_prime = [True] * (n + 1)
        is_prime[0] = is_prime[1] = False
        for i in range(2, int(n**0.5) + 1):
            if is_prime[i]:
                for j in range(i*i, n + 1, i):
                    is_prime[j] = False
        return sum(is_prime)

    number_of_primes = count_primes(n)
    number_of_non_primes = n - number_of_primes

    def factorial(n):
        result = 1
        for i in range(1, n + 1):
            result = (result * i) % modulo
        return result

    # Calculate factorial of primes, modulo-ing to avoid overflow.
    factorial_primes = factorial(number_of_primes)

    # Calculate factorial of non-primes, modulo-ing to avoid overflow.
    factorial_non_primes = factorial(number_of_non_primes)

    # Multiply the factorials, modulo-ing to avoid overflow.
    final_result = (factorial_primes * factorial_non_primes) % modulo

    return final_result

Big(O) Analysis

Time Complexity

O(n) – The dominant operation is counting the prime numbers up to n, which involves iterating from 2 to n. Within the loop, each number is checked for primality. Although primality testing can be optimized, in the general case checking up to the square root of n for each number, the primality check itself contributes a factor that is less than n over the total operation. The factorial calculations are performed a fixed number of times after the prime count is known, they do involve looping up to n, but they don't increase the complexity relative to the original primality test loop, resulting in the algorithm's overall complexity remaining O(n).

Space Complexity

O(N) – The algorithm uses a boolean array, typically named 'isPrime', of size 'N+1' to mark prime numbers. This array is created to efficiently determine prime numbers up to 'N'. Consequently, the auxiliary space used directly scales with the input 'N', resulting in a linear relationship. Therefore, the space complexity is O(N).

Edge Cases

Case	How to Handle
Input n = 0 or n = 1	Return 1 immediately as there's only one arrangement (or no arrangement to rearrange) respectively.
Large n close to the problem constraints (e.g., 100)	Use long integer type to prevent potential integer overflow when calculating factorials and apply modulo arithmetic during each multiplication.
n is a prime number itself	The count of primes might be miscalculated due to the Sieve of Eratosthenes so prime checking is important.
n close to the maximum value allowed by the constraints	Precompute and store prime numbers within the given range to avoid redundant calculations.
Prime count and composite count are significantly different (highly skewed)	The algorithm should perform optimally regardless of prime/composite number distributions within the range.
Integer overflow during factorial calculation	Apply modulo operation at each step of factorial calculation to keep the result within the required range.
Incorrect prime number calculation	Ensure a correct Sieve of Eratosthenes implementation to count primes in the specified range accurately.
Modulus operation with 0	The problem specifies a prime modulus, so division by zero during the modulus operation will never happen.

Input n = 0 or n = 1

How to Handle:

Return 1 immediately as there's only one arrangement (or no arrangement to rearrange) respectively.

Large n close to the problem constraints (e.g., 100)

How to Handle:

Use long integer type to prevent potential integer overflow when calculating factorials and apply modulo arithmetic during each multiplication.

n is a prime number itself

How to Handle:

The count of primes might be miscalculated due to the Sieve of Eratosthenes so prime checking is important.

n close to the maximum value allowed by the constraints

How to Handle:

Precompute and store prime numbers within the given range to avoid redundant calculations.

Prime count and composite count are significantly different (highly skewed)

How to Handle:

The algorithm should perform optimally regardless of prime/composite number distributions within the range.

Integer overflow during factorial calculation

How to Handle:

Apply modulo operation at each step of factorial calculation to keep the result within the required range.

Incorrect prime number calculation

How to Handle:

Ensure a correct Sieve of Eratosthenes implementation to count primes in the specified range accurately.

Modulus operation with 0

How to Handle:

The problem specifies a prime modulus, so division by zero during the modulus operation will never happen.

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the maximum possible value of `n`? I need to understand the scale for prime number calculation.
Is `n` always a positive integer?
Should the result be returned modulo a specific number? If so, what is that number?
Could you define more precisely what is meant by a 'valid arrangement'? Is it sufficient for the primes to be in prime positions, and non-primes to be in non-prime positions, or are there additional constraints?
Is there any specified behavior expected for `n` equal to 0 or 1?

Brute Force Solution

Approach

Here's how the algorithm would work step-by-step:

First, figure out all the different ways you can arrange the numbers from 1 to the input number.
For each of these arrangements, go through each position and check if a prime number is in a prime position and a non-prime number is in a non-prime position.
If, for a particular arrangement, a prime number is in a non-prime position or vice versa, then that arrangement is incorrect, and you should disregard it.
If an arrangement follows the rule where primes are in prime positions and non-primes are in non-prime positions, then count that as a valid arrangement.
After checking all possible arrangements, report the total number of valid arrangements.

Code Implementation

def prime_arrangements_brute_force(number):
    all_permutations = list(permutations(range(1, number + 1)))
    valid_arrangements_count = 0
    
    for permutation in all_permutations:
        is_valid = True
        
        for index, number_in_permutation in enumerate(permutation):
            # Check if prime is at a non-prime position or vice versa
            if is_prime(index + 1) != is_prime(number_in_permutation):

                is_valid = False
                break
        
        # Only increment count if permutation is valid
        if is_valid:
            valid_arrangements_count += 1
            
    return valid_arrangements_count % (10**9 + 7)

from itertools import permutations

def is_prime(number_to_check):
    if number_to_check <= 1:
        return False
    for i in range(2, int(number_to_check**0.5) + 1):
        if number_to_check % i == 0:
            return False
    return True

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

Here's how the algorithm would work step-by-step:

First, count how many prime numbers there are between 1 and 'n'.
Then, figure out how many non-prime numbers there are; this is just 'n' minus the number of primes.
Calculate the factorial of the number of prime numbers. This tells you how many ways you can arrange the primes in the prime positions.
Calculate the factorial of the number of non-prime numbers. This tells you how many ways you can arrange the non-primes in the non-prime positions.
Multiply these two factorials together. This gives you the total number of valid arrangements.
Since the numbers can get very big, perform the multiplication and factorial calculations using modulo to avoid exceeding the maximum number size. This keeps the result within a manageable range.

Code Implementation

def prime_arrangement(n): 
    modulo = 10**9 + 7

    def count_primes(n):
        if n < 2:
            return 0
        is_prime = [True] * (n + 1)
        is_prime[0] = is_prime[1] = False
        for i in range(2, int(n**0.5) + 1):
            if is_prime[i]:
                for j in range(i*i, n + 1, i):
                    is_prime[j] = False
        return sum(is_prime)

    number_of_primes = count_primes(n)
    number_of_non_primes = n - number_of_primes

    def factorial(n):
        result = 1
        for i in range(1, n + 1):
            result = (result * i) % modulo
        return result

    # Calculate factorial of primes, modulo-ing to avoid overflow.
    factorial_primes = factorial(number_of_primes)

    # Calculate factorial of non-primes, modulo-ing to avoid overflow.
    factorial_non_primes = factorial(number_of_non_primes)

    # Multiply the factorials, modulo-ing to avoid overflow.
    final_result = (factorial_primes * factorial_non_primes) % modulo

    return final_result

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Input n = 0 or n = 1	Return 1 immediately as there's only one arrangement (or no arrangement to rearrange) respectively.
Large n close to the problem constraints (e.g., 100)	Use long integer type to prevent potential integer overflow when calculating factorials and apply modulo arithmetic during each multiplication.
n is a prime number itself	The count of primes might be miscalculated due to the Sieve of Eratosthenes so prime checking is important.
n close to the maximum value allowed by the constraints	Precompute and store prime numbers within the given range to avoid redundant calculations.
Prime count and composite count are significantly different (highly skewed)	The algorithm should perform optimally regardless of prime/composite number distributions within the range.
Integer overflow during factorial calculation	Apply modulo operation at each step of factorial calculation to keep the result within the required range.
Incorrect prime number calculation	Ensure a correct Sieve of Eratosthenes implementation to count primes in the specified range accurately.
Modulus operation with 0	The problem specifies a prime modulus, so division by zero during the modulus operation will never happen.

Input n = 0 or n = 1

How to Handle:

Return 1 immediately as there's only one arrangement (or no arrangement to rearrange) respectively.

Large n close to the problem constraints (e.g., 100)

How to Handle:

Use long integer type to prevent potential integer overflow when calculating factorials and apply modulo arithmetic during each multiplication.

n is a prime number itself

How to Handle:

The count of primes might be miscalculated due to the Sieve of Eratosthenes so prime checking is important.

n close to the maximum value allowed by the constraints

How to Handle:

Precompute and store prime numbers within the given range to avoid redundant calculations.

Prime count and composite count are significantly different (highly skewed)

How to Handle:

The algorithm should perform optimally regardless of prime/composite number distributions within the range.

Integer overflow during factorial calculation

How to Handle:

Apply modulo operation at each step of factorial calculation to keep the result within the required range.

Incorrect prime number calculation

How to Handle:

Ensure a correct Sieve of Eratosthenes implementation to count primes in the specified range accurately.

Modulus operation with 0

How to Handle:

The problem specifies a prime modulus, so division by zero during the modulus operation will never happen.