Given a number N. The task is to find the number of ways you can draw N chords in a circle with 2*N points such that no two chords intersect. Two ways are different if there exists a chord that is present in one way and not in other. As the answer could be large print it modulo 10^9+7.
Examples:
Input : N = 2
Output : 2
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
Approach:
If we draw a chord between any two points, the current set of points gets 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) }.
The above recurrence relation is similar to the recurrence relation for nth Catalan number which is equal to 2nCn / (n+1). Instead of dividing the numeration with the denomination, multiply the numerator with the modulo inverse of the denominator as division is not allowed in the modulo domain.
Below is the implementation of the above approach:
// C++ implementation of the above approach
#include <bits/stdc++.h>
using namespace std;
// Function to calculate x^y %mod efficiently
int power(long long x, int y, int mod)
{
// Initialize the answer
long long res = 1;
while (y) {
// If power is odd
if (y & 1)
// Update the answer
res = (res * x) % mod;
// Square the base and half the exponent
x = (x * x) % mod;
y = (y >> 1);
}
// Return the value
return (int)(res % mod);
}
// Function to calculate ncr%mod efficiently
int ncr(int n, int r, int mod)
{
// Initialize the answer
long long res = 1;
// Calculate ncr in O(r)
for (int i = 1; i <= r; i += 1) {
// Multiply with the numerator factor
res = (res * (n - i + 1)) % mod;
// Calculate the inverse of factor of denominator
int inv = power(i, mod - 2, mod);
// Multiply with inverse value
res = (res * inv) % mod;
}
// Return answer value
return (int)(res%mod);
}
// Function to return the number
// of non intersecting chords
int NoOfChords(int A)
{
// define mod value
int mod = 1e9 + 7;
// Value of C(2n, n)
long long ans = ncr(2 * A, A, mod);
// Modulo inverse of (n+1)
int inv = power(A + 1, mod - 2, mod);
// Multiply with modulo inverse
ans = (ans * inv) % mod;
// Return the answer
return (int)(ans%mod);
}
// Driver code
int main()
{
int N = 2;
// Function call
cout << NoOfChords(N);
return 0;
}
// Java implementation of the approach
class GFG
{
// Function to calculate x^y %mod efficiently
static int power(long x, int y, int mod)
{
// Initialize the answer
long res = 1;
while (y != 0)
{
// If power is odd
if ((y & 1) == 1)
// Update the answer
res = (res * x) % mod;
// Square the base and half the exponent
x = (x * x) % mod;
y = (y >> 1);
}
// Return the value
return (int)(res % mod);
}
// Function to calculate ncr%mod efficiently
static int ncr(int n, int r, int mod)
{
// Initialize the answer
long res = 1;
// Calculate ncr in O(r)
for (int i = 1; i <= r; i += 1)
{
// Multiply with the numerator factor
res = (res * (n - i + 1)) % mod;
// Calculate the inverse of
// factor of denominator
int inv = power(i, mod - 2, mod);
// Multiply with inverse value
res = (res * inv) % mod;
}
// Return answer value
return (int)(res % mod);
}
// Function to return the number
// of non intersecting chords
static int NoOfChords(int A)
{
// define mod value
int mod = (int)(1e9 + 7);
// Value of C(2n, n)
long ans = ncr(2 * A, A, mod);
// Modulo inverse of (n+1)
int inv = power(A + 1, mod - 2, mod);
// Multiply with modulo inverse
ans = (ans * inv) % mod;
// Return the answer
return (int)(ans % mod);
}
// Driver code
public static void main(String[] args)
{
int N = 2;
// Function call
System.out.println(NoOfChords(N));
}
}
// This code is contributed by 29AjayKumar
# Python3 implementation of the above approach
# Function to calculate x^y %mod efficiently
def power(x, y, mod):
# Initialize the answer
res = 1
while (y):
# If power is odd
if (y & 1):
# Update the answer
res = (res * x) % mod
# Square the base and half the exponent
x = (x * x) % mod
y = (y >> 1)
# Return the value
return (res % mod)
# Function to calculate ncr%mod efficiently
def ncr(n, r, mod):
# Initialize the answer
res = 1
# Calculate ncr in O(r)
for i in range(1,r+1):
# Multiply with the numerator factor
res = (res * (n - i + 1)) % mod
# Calculate the inverse of factor of denominator
inv = power(i, mod - 2, mod)
# Multiply with inverse value
res = (res * inv) % mod
# Return answer value
return (res%mod)
# Function to return the number
# of non intersecting chords
def NoOfChords(A):
# define mod value
mod = 10**9 + 7
# Value of C(2n, n)
ans = ncr(2 * A, A, mod)
# Modulo inverse of (n+1)
inv = power(A + 1, mod - 2, mod)
# Multiply with modulo inverse
ans = (ans * inv) % mod
# Return the answer
return (ans%mod)
# Driver code
N = 2
# Function call
print(NoOfChords(N))
# This code is contributed by mohit kumar 29
// Java implementation of the above approach
using System;
class GFG
{
// Function to calculate x^y %mod efficiently
static int power(long x, int y, int mod)
{
// Initialize the answer
long res = 1;
while (y != 0)
{
// If power is odd
if ((y & 1) == 1)
// Update the answer
res = (res * x) % mod;
// Square the base and half the exponent
x = (x * x) % mod;
y = (y >> 1);
}
// Return the value
return (int)(res % mod);
}
// Function to calculate ncr%mod efficiently
static int ncr(int n, int r, int mod)
{
// Initialize the answer
long res = 1;
// Calculate ncr in O(r)
for (int i = 1; i <= r; i += 1)
{
// Multiply with the numerator factor
res = (res * (n - i + 1)) % mod;
// Calculate the inverse of factor of denominator
int inv = power(i, mod - 2, mod);
// Multiply with inverse value
res = (res * inv) % mod;
}
// Return answer value
return (int)(res % mod);
}
// Function to return the number
// of non intersecting chords
static int NoOfChords(int A)
{
// define mod value
int mod = (int)(1e9 + 7);
// Value of C(2n, n)
long ans = ncr(2 * A, A, mod);
// Modulo inverse of (n+1)
int inv = power(A + 1, mod - 2, mod);
// Multiply with modulo inverse
ans = (ans * inv) % mod;
// Return the answer
return (int)(ans % mod);
}
// Driver code
public static void Main ()
{
int N = 2;
// Function call
Console.WriteLine(NoOfChords(N));
}
}
// This code is contributed by AnkitRai01
<script>
// JavaScript implementation of the approach
// Function to calculate x^y %mod efficiently
function power(x , y , mod)
{
// Initialize the answer
var res = 1;
while (y != 0)
{
// If power is odd
if ((y & 1) == 1)
// Update the answer
res = (res * x) % mod;
// Square the base and half the exponent
x = (x * x) % mod;
y = (y >> 1);
}
// Return the value
return parseInt(res % mod);
}
// Function to calculate ncr%mod efficiently
function ncr(n , r , mod)
{
// Initialize the answer
var res = 1;
// Calculate ncr in O(r)
for (var i = 1; i <= r; i += 1)
{
// Multiply with the numerator factor
res = (res * (n - i + 1)) % mod;
// Calculate the inverse of
// factor of denominator
var inv = power(i, mod - 2, mod);
// Multiply with inverse value
res = (res * inv) % mod;
}
// Return answer value
return parseInt(res % mod);
}
// Function to return the number
// of non intersecting chords
function NoOfChords( A)
{
// define mod value
var mod = parseInt(7);
// Value of C(2n, n)
var ans = ncr(2 * A, A, mod);
// Modulo inverse of (n+1)
var inv = power(A + 1, mod - 2, mod);
// Multiply with modulo inverse
ans = (ans * inv) % mod;
// Return the answer
return parseInt(ans % mod);
}
// Driver code
var N = 2;
// Function call
document.write(NoOfChords(N));
// This code is contributed by 29AjayKumar
</script>
Output:
2
Time complexity : O(N*log(mod))
Auxiliary Space: O(1)