动态规划和一维数组的二项式系数算法

Question

I'm know a dynamic programming algorithm to compute the binomial coefficients with two-dimensional array like below . Is there any way to make use of one-dimensional array?

int binomialCoeff(int n, int k)
{
int C[n+1][k+1];
int i, j;


for (i = 0; i <= n; i++)
 {
   for (j = 0; j <= min(i, k); j++)
    {

        if (j == 0 || j == i)
            C[i][j] = 1;


        else
            C[i][j] = C[i-1][j-1] + C[i-1][j];
    }
    }

  return C[n][k];
  }

Answer 1

Your Dynamic Programming method (using 2D array) to solve Binomial Coefficient, seems correct. Note that we do not need to keep the whole table, only the prior row. So 1D implementation is possible!

Below is the code to implement it using a 1D array.

    int binomial_coefficient(int n, int k)
    {
          int C[k+1];int i,j;
          for(i=0;i<=k;i++)
                 C[i]=0;
          C[0]=1;
          for(i=1;i<=n;i++)
          {
                 for(j=min(i,k);j>0;j--)
                         C[j]=C[j]+C[j-1];
          }
          return C[k];
    }

Answer 2

Two dimensional arrays can be easily represented as one dimensional arrays.

instead of defining C[n+1][k+1] , define some C2[(n+1)*(k+1)] . Then, instead of calling C[a][b] call C2[b*(n+1)+a] .