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Prefix Factorials of a Prefix Sum Array

  • Difficulty Level : Hard
  • Last Updated : 06 Sep, 2021

Given an array arr[] consisting of N positive integers, the task is to find the prefix factorials of a prefix sum array of the given array i.e., prefix[i] = (\sum_{0}^{i}arr[i])!                 .

Examples:

Input: arr[] = {1, 2, 3, 4}
Output: 1 6 720 3628800
Explanation:
The prefix sum of the given array is {1, 3, 6, 10}. Therefore, prefix factorials of the obtained prefix sum array is {1!, (1+2)!, (1+2+3)!, (1+2+3+4)!} = {1!, 3!, 6!, 10!} = {1 6 720 3628800}.

Input: arr[] = {2, 4, 3, 1}
Output: 2 720 362880 3628800
 

Naive Approach: The simplest approach to solve the given problem is to find the prefix sum of the given array and then find the factorial of each array element in the prefix sum array. After calculating the prefix sum print the factorial array.



Below is the implementation of the above approach.

C++




// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the factorial of
// a number N
int fact(int N)
{
    // Base Case
    if (N == 1 || N == 0)
        return 1;
 
    // Find the factorial recursively
    return N * fact(N - 1);
}
 
// Function to find the prefix
// factorial array
void prefixFactorialArray(int arr,
                          int N)
{
    // Find the prefix sum array
    for (int i = 1; i < N; i++) {
        arr[i] += arr[i - 1];
    }
 
    // Find the factorials of each
    // array element
    for (int i = 0; i < N; i++) {
        arr[i] = fact(arr[i]);
    }
 
    // Print the resultant array
    for (int i = 0; i < N; i++) {
        cout << arr[i] << " ";
    }
}
 
// Driver Code
int main()
{
    int arr[] = { 1, 2, 3, 4 };
    int N = sizeof(arr) / sizeof(arr[0]);
    prefixFactorialArray(arr, N);
 
    return 0;
}

Java




// Java program for the above approach
class GFG{
 
// Function to find the factorial of
// a number N
static int fact(int N)
{
     
    // Base Case
    if (N == 1 || N == 0)
        return 1;
 
    // Find the factorial recursively
    return N * fact(N - 1);
}
 
// Function to find the prefix
// factorial array
static void prefixFactorialArray(int[] arr, int N)
{
 
    // Find the prefix sum array
    for(int i = 1; i < N; i++)
    {
        arr[i] += arr[i - 1];
    }
 
    // Find the factorials of each
    // array element
    for(int i = 0; i < N; i++)
    {
        arr[i] = fact(arr[i]);
    }
 
    // Print the resultant array
    for(int i = 0; i < N; i++)
    {
        System.out.print(arr[i] + " ");
    }
}
 
// Driver Code
public static void main(String[] args)
{
    int[] arr = { 1, 2, 3, 4 };
    int N = arr.length;
 
    prefixFactorialArray(arr, N);
}
}
 
// This code is contributed by ukasp

Python3




# Python implementation of the approach
def fact(N):
   
    # Base Case
    if (N == 1 or N == 0):
        return 1
 
    # Find the factorial recursively
    return N * fact(N - 1)
 
 
# Function to find the prefix
# factorial array
def prefixFactorialArray(arr, N):
 
    # Find the prefix sum array
    for i in range(1, N):
        arr[i] += arr[i - 1]
 
    # Find the factorials of each
    # array element
    for i in range(N):
        arr[i] = fact(arr[i])
 
        # Print the resultant array
    for i in range(N):
        print(arr[i], end=" ")
 
# Driver Code
if __name__ == "__main__":
 
    arr = [1, 2, 3, 4]
    N = len(arr)
    prefixFactorialArray(arr, N)
 
# This code is contributed by kirtishsurangalikar

C#




// C# program for the above approach
using System;
         
class GFG{
 
// Function to find the factorial of
// a number N
static int fact(int N)
{
     
    // Base Case
    if (N == 1 || N == 0)
        return 1;
 
    // Find the factorial recursively
    return N * fact(N - 1);
}
 
// Function to find the prefix
// factorial array
static void prefixFactorialArray(int[] arr,
                                 int N)
{
     
    // Find the prefix sum array
    for(int i = 1; i < N; i++)
    {
        arr[i] += arr[i - 1];
    }
 
    // Find the factorials of each
    // array element
    for(int i = 0; i < N; i++)
    {
        arr[i] = fact(arr[i]);
    }
 
    // Print the resultant array
    for(int i = 0; i < N; i++)
    {
        Console.Write(arr[i] + " ");
    }
}
     
// Driver Code
public static void Main()
{
    int[] arr = { 1, 2, 3, 4 };
    int N = arr.Length;
     
    prefixFactorialArray(arr, N);
}
}
 
// This code is contributed by code_hunt

Javascript




<script>
 
// JavaScript program for the above approach
 
 
// Function to find the factorial of
// a number N
function fact(N) {
    // Base Case
    if (N == 1 || N == 0)
        return 1;
 
    // Find the factorial recursively
    return N * fact(N - 1);
}
 
// Function to find the prefix
// factorial array
function prefixFactorialArray(arr, N) {
    // Find the prefix sum array
    for (let i = 1; i < N; i++) {
        arr[i] += arr[i - 1];
    }
 
    // Find the factorials of each
    // array element
    for (let i = 0; i < N; i++) {
        arr[i] = fact(arr[i]);
    }
 
    // Print the resultant array
    for (let i = 0; i < N; i++) {
        document.write(arr[i] + " ");
    }
}
 
// Driver Code
 
let arr = [1, 2, 3, 4];
let N = arr.length;
prefixFactorialArray(arr, N);
 
</script>

Output:

1 6 720 3628800 

Time Complexity: O(N*M), where M is the sum of the array elements
Auxiliary Space: O(1)

Efficient Approach: The above approach can also be optimized by precalculating the factorial of sum of the array elements so that the factorial calculation at each index is can be calculated in O(1) time.

Below is an implementation of the above approach:

C++




// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the factorial of
// prefix sum at every possible index
void prefixFactorialArray(int A[], int N)
{
    // Find the prefix sum array
    for (int i = 1; i < N; i++) {
        A[i] += A[i - 1];
    }
 
    // Stores the factorial of all the
    // element till the sum of array
    int fact[A[N - 1] + 1];
    fact[0] = 1;
 
    // Find the factorial array
    for (int i = 1; i <= A[N - 1]; i++) {
        fact[i] = i * fact[i - 1];
    }
 
    // Find the factorials of
    // each array element
    for (int i = 0; i < N; i++) {
        A[i] = fact[A[i]];
    }
 
    // Print the resultant array
    for (int i = 0; i < N; i++) {
        cout << A[i] << " ";
    }
}
 
// Driver Code
int main()
{
    int arr[] = { 1, 2, 3, 4 };
    int N = sizeof(arr) / sizeof(arr[0]);
    prefixFactorialArray(arr, N);
 
    return 0;
}

Java




// Java program for the above approach
class GFG{
 
// Function to find the factorial of
// prefix sum at every possible index
static void prefixFactorialArray(int A[], int N)
{
     
    // Find the prefix sum array
    for(int i = 1; i < N; i++)
    {
        A[i] += A[i - 1];
    }
 
    // Stores the factorial of all the
    // element till the sum of array
    int fact[] = new int[A[N - 1] + 1];
    fact[0] = 1;
 
    // Find the factorial array
    for(int i = 1; i <= A[N - 1]; i++)
    {
        fact[i] = i * fact[i - 1];
    }
 
    // Find the factorials of
    // each array element
    for(int i = 0; i < N; i++)
    {
        A[i] = fact[A[i]];
    }
 
    // Print the resultant array
    for(int i = 0; i < N; i++)
    {
        System.out.print(A[i] + " ");
    }
}
 
// Driver code
public static void main(String[] args)
{
    int arr[] = { 1, 2, 3, 4 };
    int N = arr.length;
     
    prefixFactorialArray(arr, N);
}
}
 
// This code is contributed by abhinavjain194

Python3




# // python program for the above approach
 
# // Function to find the factorial of
# // prefix sum at every possible index
def prefixFactorialArray(A, N):
   
    # // Find the prefix sum array
    for i in range(1, N):
        A[i] += A[i - 1]
 
    # // Stores the factorial of all the
    # // element till the sum of array
    fact = [0 for x in range(A[N - 1] + 1)]
    fact[0] = 1
 
    # // Find the factorial array
    for i in range(1, A[N-1]+1):
        fact[i] = i * fact[i - 1]
 
    # // Find the factorials of
    # // each array element
    for i in range(0, N):
        A[i] = fact[A[i]]
         
    # // Print the resultant array
    for i in range(0, N):
        print(A[i], end=" ")
 
# Driver code
arr = [1, 2, 3, 4]
N = len(arr)
prefixFactorialArray(arr, N)
 
# This code is contributed by amreshkumar3.

C#




// C# program for the above approach
using System;
class GFG {
     
    // Function to find the factorial of
    // prefix sum at every possible index
    static void prefixFactorialArray(int[] A, int N)
    {
          
        // Find the prefix sum array
        for(int i = 1; i < N; i++)
        {
            A[i] += A[i - 1];
        }
      
        // Stores the factorial of all the
        // element till the sum of array
        int[] fact = new int[A[N - 1] + 1];
        fact[0] = 1;
      
        // Find the factorial array
        for(int i = 1; i <= A[N - 1]; i++)
        {
            fact[i] = i * fact[i - 1];
        }
      
        // Find the factorials of
        // each array element
        for(int i = 0; i < N; i++)
        {
            A[i] = fact[A[i]];
        }
      
        // Print the resultant array
        for(int i = 0; i < N; i++)
        {
            Console.Write(A[i] + " ");
        }
    }
 
  // Driver code
  static void Main() {
    int[] arr = { 1, 2, 3, 4 };
    int N = arr.Length;
      
    prefixFactorialArray(arr, N);
  }
}
 
// This code is contributed by divyeshrabadiya07.

Javascript




<script>
 
// Javascript program for the above approach
 
 
// Function to find the factorial of
// prefix sum at every possible index
function prefixFactorialArray(A, N) {
    // Find the prefix sum array
    for (let i = 1; i < N; i++) {
        A[i] += A[i - 1];
    }
 
    // Stores the factorial of all the
    // element till the sum of array
    let fact = new Array(A[N - 1] + 1);
    fact[0] = 1;
 
    // Find the factorial array
    for (let i = 1; i <= A[N - 1]; i++) {
        fact[i] = i * fact[i - 1];
    }
 
    // Find the factorials of
    // each array element
    for (let i = 0; i < N; i++) {
        A[i] = fact[A[i]];
    }
 
    // Print the resultant array
    for (let i = 0; i < N; i++) {
        document.write(A[i] + " ");
    }
}
 
// Driver Code
 
let arr = [1, 2, 3, 4];
let N = arr.length
prefixFactorialArray(arr, N);
 
// This code is contributed by _saurabh_jaiswal.
</script>
Output: 
1 6 720 3628800

 

Time Complexity: O(N + M), where M is the sum of the array elements.
Auxiliary Space: O(M), where M is the sum of the array elements.

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