Count ways to divide circle using N non-intersecting chords

Given a number N, find the number of ways you can draw N chords in a circle with 2*N points such that no 2 chords intersect.
Two ways are different if there exists a chord which is present in one way and not in other.

Examples:

Input : N = 2
Output : 2
Explanation: If points are numbered 1 to 4 in 
clockwise direction, then different ways to 
draw chords are:
{(1-2), (3-4)} and {(1-4), (2-3)}


Input : N = 1
Output : 1
Explanation: Draw a chord between points 1 and 2.



If we draw a chord between any two points, can you observe the current set of points getting broken into two smaller sets S_1 and S_2. If we draw a chord from a point in S_1 to a point in S_2, it will surely intersect the chord we’ve just drawn.
So, we can arrive at a recurrence that Ways(n) = sum[i = 0 to n-1] { Ways(i)*Ways(n-i-1) }.
Here we iterate over i, assuming that size of one of the sets is i and size of another set automatically is (n-i-1) since we’ve already used a pair of points and i pair of points in one set.

C++


// cpp code to count ways 
// to divide circle using
// N non-intersecting chords.
#include <bits/stdc++.h>
using namespace std;

int chordCnt( int A){

    // n = no of points required
    int n = 2 * A;
    
    // dp array containing the sum
    int dpArray[n + 1]={ 0 };
    dpArray[0] = 1;
    dpArray[2] = 1;
    for (int i=4;i<=n;i+=2){
        for (int j=0;j<i-1;j+=2){ 
            
          dpArray[i] +=
            (dpArray[j]*dpArray[i-2-j]);
        }
    } 

    // returning the required number
    return dpArray[n];
}
// Driver function
int main()
{

    int N;
    N = 2;
cout<<chordCnt( N)<<'\n';
    N = 1;
cout<<chordCnt( N)<<'\n';
    N = 4;
cout<<chordCnt( N)<<'\n';
    return 0;
}

// This code is contributed by Gitanjali.

Java

// Java code to count ways
// to divide circle using
// N non-intersecting chords.
import java.io.*;

class GFG {
    static int chordCnt(int A)
    {

        // n = no of points required
        int n = 2 * A;

        // dp array containing the sum
        int[] dpArray = new int[n + 1];
        dpArray[0] = 1;
        dpArray[2] = 1;
        for (int i = 4; i <= n; i += 2) {
            for (int j = 0; j < i - 1; j += 2) 
            {
                dpArray[i] += (dpArray[j] * 
                              dpArray[i - 2 - j]);
            }
        }

        // returning the required number
        return dpArray[n];
    }
    public static void main(String[] args)
    {
        int N;
        N = 2;
        System.out.println(chordCnt(N));
        N = 1;
        System.out.println(chordCnt(N));
        N = 4;
        System.out.println(chordCnt(N));
    }
}

// This code is contributed by Gitanjali.

Python 3

# python code to count ways to divide
# circle using N non-intersecting chords.
def chordCnt( A):

    # n = no of points required
    n = 2 * A

    # dp array containing the sum
    dpArray = [0]*(n + 1)
    dpArray[0] = 1
    dpArray[2] = 1
    for i in range(4, n + 1, 2):
        for j in range(0, i-1, 2):
            dpArray[i] += (dpArray[j]*dpArray[i-2-j])

    # returning the required number
    return int(dpArray[n])

# driver code
N = 2
print(chordCnt( N))
N = 1
print(chordCnt( N))
N = 4
print(chordCnt( N))

C#

// C# code to count ways to divide 
// circle using N non-intersecting chords.
using System;

class GFG {
    
    static int chordCnt(int A)
    {
        // n = no of points required
        int n = 2 * A;

        // dp array containing the sum
        int[] dpArray = new int[n + 1];
        dpArray[0] = 1;
        dpArray[2] = 1;
        
        for (int i = 4; i <= n; i += 2) 
        {
            for (int j = 0; j < i - 1; j += 2)
            {
                dpArray[i] += (dpArray[j] * dpArray[i - 2 - j]);
            }
        }

        // returning the required number
        return dpArray[n];
    }
    
    // Driver code
    public static void Main()
    {
        int N;
        N = 2;
        Console.WriteLine(chordCnt(N));
        N = 1;
        Console.WriteLine(chordCnt(N));
        N = 4;
        Console.WriteLine(chordCnt(N));
    }
}

// This code is contributed by vt_m.

Output:

2
1
14


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