Count permutations of all integers upto N that can form an acyclic graph based on given conditions

Given an integer N, the task is to find the number of permutations of integers from the range [1, N] that can form an acyclic graph according to the following conditions:

• For every 1 ? i ? N, find the largest j such that 1 ? j < i and A[j] > A[i], and add an undirected edge between node i and node j.
• For every 1 ? i ? N, find the smallest j such that i < j ? N and A[j] > A[i], and add an undirected edge between node i and node j.

Examples:

Input: 3
Output: 4
Explanation:
{1, 2, 3}: Possible (Graph Edges : { [1, 2], [2, 3] }. Therefore, no cycle exists) 
{1, 3, 2}: Possible (Graph Edges : { [1, 3], [2, 3] }. Therefore, no cycle exists) 
{2, 1, 3}: Not possible (Graph Edges : { [2, 3], [1, 3], [1, 2] }. Therefore, cycle exists) 
{2, 3, 1}: Possible (Graph Edges : { [2, 3], [1, 3] }. Therefore, no cycle exists) 
{3, 1, 2}: Not possible (Graph Edges : { [1, 2], [1, 3], [2, 3] }. Therefore, cycle exists) 
{3, 2, 1}: Possible (Graph Edges : { [2, 3], [1, 2] }. Therefore, no cycle exists)

Input : 4
Output : 8

Approach: Follow the steps below to solve the problem:

• There are a total of N! possible arrays that can be generated consisting of permutations of integers from the range [1, N].
• According to the given conditions, if i, j, k (i < j < k) are indices from the array, then if A[i] > A[j] and A[j] < A[k], then there will be a cycle consisting of edges {[A[j], A[i]], [A[j], A[k]], [A[i], A[k]]}.
• Removing these permutations, there are 2^{(N – 1)} possible permutations remaining.
• Remaining permutations gives only two possible positions for integers from the range [1, N – 1] and 1 possible position for N.

Illustration:
For N = 3: 
Permutation {2, 1, 3} contains a cycle as A[0] > A[1] and A[1] < A[2]. 
Permutation {3, 1, 2} contains a cycle as A[0] > A[1] and A[1] < A[2].
All remaining permutations can generate an acyclic graph. 
Therefore, the count of valid permutations = 3! – 2 = 4 = 2^{3 – 1}

• Therefore, print 2^{N – 1} as the required answer.

Below is the implementation of the above approach : 

C++

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

// Find the count of possible graphs
void possibleAcyclicGraph(int N)
{
    cout << pow(2, N - 1);
    return;
}

// Driver Code
int main()
{

    int N = 4;
    possibleAcyclicGraph(N);

    return 0;
}

Java

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

// Find the count of
// possible graphs
static void possibleAcyclicGraph(int N)
{
  System.out.print((int)Math.pow(2,
                                 N - 1));
  return;
}

// Driver Code
public static void main(String[] args)
{
  int N = 4;
  possibleAcyclicGraph(N);
}
}

// This code is contributed by Princi Singh

Python3

# Python3 implementation of above approach

# Find the count of possible graphs
def possibleAcyclicGraph(N):

    print(pow(2, N - 1))
    return

# Driver code
if __name__ == '__main__':

    N = 4

    possibleAcyclicGraph(N)

# This code is contributed by mohit kumar 29

C#

// C# implementation of
// the above approach
using System;

class GFG{

// Find the count of
// possible graphs
static void possibleAcyclicGraph(int N)
{
    Console.Write((int)Math.Pow(2, N - 1));

    return;
}

// Driver Code
public static void Main()
{
    int N = 4;

    possibleAcyclicGraph(N);
}
}

// This code is contributed by sanjoy_62

Javascript

<script>

// Javascript implementation of above approach

// Find the count of
// possible graphs
function possibleAcyclicGraph(N)
{
    document.write(Math.pow(2, N - 1));
    return;
}

// Driver Code
let N = 4;

possibleAcyclicGraph(N);

// This code is contributed by target_2

</script>

Output

8

Time Complexity: O(logN)
Space Complexity: O(1)