USACO 3.2 Stringsobits

USACO 3.2 Stringsobits

首先计算出组合数。用cmb_num[i][j]表示i位数中，"1的位数小于等于j"的数的个数。
这样，我们从最左边开始，如果cmb_num[i-1][j]的数大于n，说明第一位为0，因为用i-1位数中"1的位数小于等于j"的数已经大于n个了。
如果小于n，说明第一位为1，需要i位，才能使"1的位数小于等于j"的数大于n个了。既然第一位已经是1了，接下来的i-1位组成的数的1的个数只能小于等于n-1位了。迭代输出每一位即可。
只是要注意溢出的问题以及cmb_num[0][1]。

#include  < iostream >
#include  < fstream >

using   namespace  std;

ifstream fin( " kimbits.in " );
ofstream fout( " kimbits.out " );

#ifdef _DEBUG
#define  out cout
#define  in cin
#else
#define  out fout
#define  in fin
#endif

unsigned  int  cmb_num[ 32 ][ 32 ];

void  build_cmb_num()
{
     for ( int  i = 0 ;i < 32 ; ++ i)
        cmb_num[i][ 0 ]  =   1 ;

     for ( int  i = 1 ;i < 32 ; ++ i)
         for ( int  j = 1 ;j <= i; ++ j)
            cmb_num[i][j]  =  cmb_num[i - 1 ][j - 1 ] + cmb_num[i - 1 ][j];

     for ( int  i = 0 ;i < 32 ; ++ i)
         for ( int  j = 1 ;j < 32 ; ++ j){
            cmb_num[i][j] += cmb_num[i][j - 1 ];
        }
}


void  solve()
{
    build_cmb_num();

    unsigned  n,l,i;
     in >> n >> l >> i;

     for (unsigned idx = n;idx > 0 ; -- idx){
         if ( i >  cmb_num[idx - 1 ][l] ){
             out << 1 ;
            i -= cmb_num[idx - 1 ][l];
            l -- ;
        } else {
             out << 0 ;
        }   
    }

     out << endl;
}

int  main( int  argc, char   * argv[])
{
    solve(); 
     return   0 ;
}

附题：
Stringsobits
Kim Schrijvers

Consider an ordered set S of strings of N (1 <= N <= 31) bits. Bits, of course, are either 0 or 1.

This set of strings is interesting because it is ordered and contains all possible strings of length N that have L (1 <= L <= N) or fewer bits that are `1'.

Your task is to read a number I (1 <= I <= sizeof(S)) from the input and print the Ith element of the ordered set for N bits with no more than L bits that are `1'.

PROGRAM NAME: kimbits

INPUT FORMAT

A single line with three space separated integers: N, L, and I.

SAMPLE INPUT (file kimbits.in)

5 3 19

OUTPUT FORMAT

A single line containing the integer that represents the Ith element from the order set, as described.

SAMPLE OUTPUT (file kimbits.out)

10011