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.
// 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 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 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# 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. |
<?php // PHP code to count ways // to divide circle using // N non-intersecting chords. function chordCnt( $A )
{ // n = no of points required
$n = 2 * $A ;
// dp array containing the sum
$dpArray = array_fill (0, $n + 1, 0);
$dpArray [0] = 1;
$dpArray [2] = 1;
for ( $i = 4; $i <= $n ; $i += 2)
{
for ( $j = 0; $j < $i - 1; $j += 2)
{
$dpArray [ $i ] += ( $dpArray [ $j ] *
$dpArray [ $i - 2 - $j ]);
}
}
// returning the required number
return $dpArray [ $n ];
} // Driver Code $N = 2;
echo chordCnt( $N ), "\n" ;
$N = 1;
echo chordCnt( $N ), "\n" ;
$N = 4;
echo chordCnt( $N ), "\n" ;
// This code is contributed by Ryuga ?> |
<script> // JavaScript code to count ways // to divide circle using // N non-intersecting chords. function chordCnt( A){
// n = no of points required
var n = 2 * A;
// dp array containing the sum
var dpArray = Array(n+1).fill(0);
dpArray[0] = 1;
dpArray[2] = 1;
for ( var i=4;i<=n;i+=2){
for ( var j=0;j<i-1;j+=2){
dpArray[i] +=
(dpArray[j]*dpArray[i-2-j]);
}
}
// returning the required number
return dpArray[n];
} // Driver function var N;
N = 2; document.write( chordCnt( N) + '<br>' );
N = 1; document.write( chordCnt( N) + '<br>' );
N = 4; document.write( chordCnt( N) + '<br>' );
</script> |
Output:
2 1 14
Time Complexity: O(n2)
Auxiliary Space: O(n)
Please suggest if someone has a better solution which is more efficient in terms of space and time.