Approach 1:
C(n,r) = n!/(n-r)!r!
Approach 2:
In the book Combinatorial Algorithms by wilf, i have found this:
C(n,r) can be written as C(n-1,r) + C(n-1,r-1)
.
e.g.
C(7,4) = C(6,4) + C(6,3)
= C(5,4) + C(5,3) + C(5,3) + C(5,2)
. .
. .
. .
. .
After solving
= C(4,4) + C(4,1) + 3*C(3,3) + 3*C(3,1) + 6*C(2,1) + 6*C(2,2)
As you can see, the final solution doesn't need any multiplication. In every form C(n,r), either n==r or r==1.
Here is the sample code i have implemented:
int foo(int n,int r)
{
if(n==r) return 1;
if(r==1) return n;
return foo(n-1,r) + foo(n-1,r-1);
}
See output here.
In the approach 2, there are overlapping sub-problems where we are calling recursion to solve the same sub-problems again. We can avoid it by using Dynamic Programming.
I want to know which is the better way to calculate C(n,r)?.
Both approaches will save time, but the first one is very prone to integer overflow.
Approach 1:
This approach will generate result in shortest time (in at most n/2
iterations), and the possibility of overflow can be reduced by doing the multiplications carefully:
long long C(int n, int r) {
if(r > n - r) r = n - r; // because C(n, r) == C(n, n - r)
long long ans = 1;
int i;
for(i = 1; i <= r; i++) {
ans *= n - r + i;
ans /= i;
}
return ans;
}
This code will start multiplication of the numerator from the smaller end, and as the product of any k
consecutive integers is divisible by k!
, there will be no divisibility problem. But the possibility of overflow is still there, another useful trick may be dividing n - r + i
and i
by their GCD before doing the multiplication and division (and still overflow may occur).
Approach 2:
In this approach, you'll be actually building up the Pascal's Triangle. The dynamic approach is much faster than the recursive one (the first one is O(n^2)
while the other is exponential). However, you'll need to use O(n^2)
memory too.
# define MAX 100 // assuming we need first 100 rows
long long triangle[MAX + 1][MAX + 1];
void makeTriangle() {
int i, j;
// initialize the first row
triangle[0][0] = 1; // C(0, 0) = 1
for(i = 1; i < MAX; i++) {
triangle[i][0] = 1; // C(i, 0) = 1
for(j = 1; j <= i; j++) {
triangle[i][j] = triangle[i - 1][j - 1] + triangle[i - 1][j];
}
}
}
long long C(int n, int r) {
return triangle[n][r];
}
Then you can look up any C(n, r)
in O(1)
time.
If you need a particular C(n, r)
(i.e. the full triangle is not needed), then the memory consumption can be made O(n)
by overwriting the same row of the triangle, top to bottom.
# define MAX 100
long long row[MAX + 1];
int C(int n, int r) {
int i, j;
// initialize by the first row
row[0] = 1; // this is the value of C(0, 0)
for(i = 1; i <= n; i++) {
for(j = i; j > 0; j--) {
// from the recurrence C(n, r) = C(n - 1, r - 1) + C(n - 1, r)
row[j] += row[j - 1];
}
}
return row[r];
}
The inner loop is started from the end to simplify the calculations. If you start it from index 0, you'll need another variable to store the value being overwritten.