Number of possible permutations when absolute difference between number of elements to the right and left are given

Given an array of N elements where each element i, the absolute difference between total elements to the right and left of it are given. Find the number of possible ordering of the actual array elements.
Examples:

Input : N = 5, arr[] = {2, 4, 4, 0, 2} 
Output : 4 
There are four possible orders, as follows: 
2, 1, 4, 5, 3 
2, 5, 4, 1, 3 
3, 1, 4, 5, 2 
3, 5, 4, 1, 2
Input : N = 7, arr[] = {6, 4, 0, 2, 4, 0, 2} 
Output : 0 
No any valid order is possible hence answer is 0. 

Approach: Divide the problem into two parts. When N is odd and when N is even. 

• Case 1: When N is odd. 
  Consider N = 7, there are 7 empty spaces and the absolute difference between the elements to the left and right must be like [6 4 2 0 2 4 6]. Observe that the element which is at the middle must have absolute difference 0, while other elements are from 2 to N-1 and each of their counts should be 2. If it doesn’t fulfill it then there is no valid order else for each element i from 2 to N-1 we have 2 ways to fill the spaces, hence total ways will be the product of all the ways. 
  
• Case 2: When N is even. 
  Consider N = 6, There are 6 spaces and it will be like [5 3 1 1 3 5], where a[i] gives the absolute difference between the number of elements to the left and right. For each a[i] we have 2 ways, hence answer will be the product of all the ways. 
  

Below is the implementation of the approach: 

C++

// C++ implementation of the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find the number of permutations
// possible of the original array to satisfy
// the given absolute differences
int totalways(int* arr, int n)
{
    // To store the count of each
    // a[i] in a map
    unordered_map<int, int> cnt;
    for (int i = 0; i < n; ++i) {
        cnt[arr[i]]++;
    }

    // if n is odd
    if (n % 2 == 1) {
        int start = 0, endd = n - 1;

        // check the count of each whether
        // it satisfy the given criteria or not
        for (int i = start; i <= endd; i = i + 2) {
            if (i == 0) {

                // there is only 1 way
                // for middle element.
                if (cnt[i] != 1) {
                    return 0;
                }
            }
            else {

                // for others there are 2 ways.
                if (cnt[i] != 2) {
                    return 0;
                }
            }
        }

        // now find total ways
        int ways = 1;
        start = 2, endd = n - 1;
        for (int i = start; i <= endd; i = i + 2) {
            ways = ways * 2;
        }
        return ways;
    }

    // When n is even.
    else if (n % 2 == 0) {

        // there will be no middle element so
        // for each a[i] there will be 2 ways
        int start = 1, endd = n - 1;
        for (int i = 1; i <= endd; i = i + 2) {
            if (cnt[i] != 2)
                return 0;
        }
        int ways = 1;
        for (int i = start; i <= endd; i = i + 2) {
            ways = ways * 2;
        }
        return ways;
    }
}

// Driver Code
int main()
{
    int N = 5;

    int arr[N] = { 2, 4, 4, 0, 2 };

    cout<<totalways(arr, N);

    return 0;
}

Java

// Java implementation of the above approach
import java.util.*;

class GFG
{

// Function to find the number of permutations
// possible of the original array to satisfy
// the given absolute differences
static int totalways(int[] arr, int n)
{
    // To store the count of each
    // a[i] in a map
    HashMap<Integer,
            Integer>cnt = new HashMap<Integer,
                                      Integer>();

    for (int i = 0 ; i < n; i++)
    {
        if(cnt.containsKey(arr[i]))
        {
            cnt.put(arr[i], cnt.get(arr[i])+1);
        }
        else
        {
            cnt.put(arr[i], 1);
        }
    }

    // if n is odd
    if (n % 2 == 1)
    {
        int start = 0, endd = n - 1;

        // check the count of each whether
        // it satisfy the given criteria or not
        for (int i = start; i <= endd; i = i + 2)
        {
            if (i == 0)
            {

                // there is only 1 way
                // for middle element.
                if (cnt.get(i) != 1)
                {
                    return 0;
                }
            }
            else
            {

                // for others there are 2 ways.
                if (cnt.get(i) != 2)
                {
                    return 0;
                }
            }
        }

        // now find total ways
        int ways = 1;
        start = 2; endd = n - 1;
        for (int i = start; i <= endd; i = i + 2)
        {
            ways = ways * 2;
        }
        return ways;
    }

    // When n is even.
    else if (n % 2 == 0)
    {

        // there will be no middle element so
        // for each a[i] there will be 2 ways
        int start = 1, endd = n - 1;
        for (int i = 1; i <= endd; i = i + 2)
        {
            if (cnt.get(i) != 2)
                return 0;
        }
        int ways = 1;
        for (int i = start; i <= endd; i = i + 2)
        {
            ways = ways * 2;
        }
        return ways;
    }
    return Integer.MIN_VALUE;
}

// Driver Code
public static void main(String[] args)
{
    int N = 5;

    int []arr = { 2, 4, 4, 0, 2 };

    System.out.println(totalways(arr, N));
}
}

// This code is contributed by Princi Singh

Python3

# Python3 implementation of the above approach

# Function to find the number of permutations
# possible of the original array to satisfy
# the given absolute differences
def totalways(arr, n):

    # To store the count of each
    # a[i] in a map
    cnt = dict()
    for i in range(n):
        cnt[arr[i]] = cnt.get(arr[i], 0) + 1

    # if n is odd
    if (n % 2 == 1):
        start, endd = 0, n - 1

        # check the count of each whether
        # it satisfy the given criteria or not
        for i in range(start, endd + 1, 2):
            if (i == 0):

                # there is only 1 way
                # for middle element.
                if (cnt[i] != 1):
                    return 0
            else:

                # for others there are 2 ways.
                if (cnt[i] != 2):
                    return 0

        # now find total ways
        ways = 1
        start = 2
        endd = n - 1
        for i in range(start, endd + 1, 2):
            ways = ways * 2
        return ways

    # When n is even.
    elif (n % 2 == 0):

        # there will be no middle element so
        # for each a[i] there will be 2 ways
        start = 1
        endd = n - 1
        for i in range(1, endd + 1, 2):
            if (cnt[i] != 2):
                return 0
        ways = 1
        for i in range(start, endd + 1, 2):
            ways = ways * 2
        return ways

# Driver Code
N = 5

arr = [2, 4, 4, 0, 2 ]

print(totalways(arr, N))

# This code is contributed by Mohit Kumar

C#

// C# implementation of the above approach
using System;
using System.Collections.Generic;

class GFG
{

// Function to find the number of permutations
// possible of the original array to satisfy
// the given absolute differences
static int totalways(int[] arr, int n)
{
    // To store the count of each
    // a[i] in a map
    Dictionary<int,
               int> cnt = new Dictionary<int,
                                         int>();

    for (int i = 0 ; i < n; i++)
    {
        if(cnt.ContainsKey(arr[i]))
        {
            cnt[arr[i]] = cnt[arr[i]] + 1;
        }
        else
        {
            cnt.Add(arr[i], 1);
        }
    }

    // if n is odd
    if (n % 2 == 1)
    {
        int start = 0, endd = n - 1;

        // check the count of each whether
        // it satisfy the given criteria or not
        for (int i = start; i <= endd; i = i + 2)
        {
            if (i == 0)
            {

                // there is only 1 way
                // for middle element.
                if (cnt[i] != 1)
                {
                    return 0;
                }
            }
            else
            {

                // for others there are 2 ways.
                if (cnt[i] != 2)
                {
                    return 0;
                }
            }
        }

        // now find total ways
        int ways = 1;
        start = 2; endd = n - 1;
        for (int i = start; i <= endd; i = i + 2)
        {
            ways = ways * 2;
        }
        return ways;
    }

    // When n is even.
    else if (n % 2 == 0)
    {

        // there will be no middle element so
        // for each a[i] there will be 2 ways
        int start = 1, endd = n - 1;
        for (int i = 1; i <= endd; i = i + 2)
        {
            if (cnt[i] != 2)
                return 0;
        }

        int ways = 1;
        for (int i = start; i <= endd; i = i + 2)
        {
            ways = ways * 2;
        }
        return ways;
    }
    return int.MinValue;
}

// Driver Code
public static void Main(String[] args)
{
    int N = 5;

    int []arr = { 2, 4, 4, 0, 2 };

    Console.WriteLine(totalways(arr, N));
}
}

// This code is contributed by 29AjayKumar

Javascript

<script>

      // JavaScript implementation of the above approach
      // Function to find the number of permutations
      // possible of the original array to satisfy
      // the given absolute differences
      function totalways(arr, n) {
        // To store the count of each
        // a[i] in a map
        var cnt = {};

        for (var i = 0; i < n; i++) {
          if (cnt.hasOwnProperty(arr[i])) {
            cnt[arr[i]] = cnt[arr[i]] + 1;
          } else {
            cnt[arr[i]] = 1;
          }
        }

        // if n is odd
        if (n % 2 === 1) {
          var start = 0,
            endd = n - 1;

          // check the count of each whether
          // it satisfy the given criteria or not
          for (var i = start; i <= endd; i = i + 2) {
            if (i === 0) {
              // there is only 1 way
              // for middle element.
              if (cnt[i] !== 1) {
                return 0;
              }
            } else {
              // for others there are 2 ways.
              if (cnt[i] !== 2) {
                return 0;
              }
            }
          }

          // now find total ways
          var ways = 1;
          start = 2;
          endd = n - 1;
          for (var i = start; i <= endd; i = i + 2) {
            ways = ways * 2;
          }
          return ways;
        }

        // When n is even.
        else if (n % 2 === 0) {
          // there will be no middle element so
          // for each a[i] there will be 2 ways
          var start = 1,
            endd = n - 1;
          for (var i = 1; i <= endd; i = i + 2) {
            if (cnt[i] !== 2) return 0;
          }

          var ways = 1;
          for (var i = start; i <= endd; i = i + 2) {
            ways = ways * 2;
          }
          return ways;
        }
        return -2147483648;
      }

      // Driver Code
      var N = 5;
      var arr = [2, 4, 4, 0, 2];

      document.write(totalways(arr, N));

</script>

Output:

4

Time Complexity: O(N), where N is the size of the given array.
Auxiliary Space: O(N) for using a hashmap to store the frequency of the given elements.