Question

What’s the best algorithm to find all binary strings of length n that contain k bits set? For example, if n=4 and k=3, there are…

0111
1011
1101
1110

I need a good way to generate these given any n and any k so I’d prefer it to be done with strings.

Answer 1

This method will generate all integers with exactly N ‘1’ bits.

From https://graphics.stanford.edu/~seander/bithacks.html#NextBitPermutation

Compute the lexicographically next bit permutation

Suppose we have a pattern of N bits set to 1 in an integer and we want
the next permutation of N 1 bits in a lexicographical sense. For
example, if N is 3 and the bit pattern is 00010011, the next patterns
would be 00010101, 00010110, 00011001, 00011010, 00011100, 00100011,
and so forth. The following is a fast way to compute the next
permutation.

unsigned int v; // current permutation of bits
unsigned int w; // next permutation of bits

unsigned int t = v | (v - 1); // t gets v's least significant 0 bits set to 1
// Next set to 1 the most significant bit to change,
// set to 0 the least significant ones, and add the necessary 1 bits.
w = (t + 1) | (((~t & -~t) - 1) >> (__builtin_ctz(v) + 1));

The __builtin_ctz(v) GNU C compiler intrinsic for x86 CPUs returns the number of trailing zeros. If you are using Microsoft compilers for
x86, the intrinsic is _BitScanForward. These both emit a bsf
instruction, but equivalents may be available for other architectures.
If not, then consider using one of the methods for counting the
consecutive zero bits mentioned earlier. Here is another version that
tends to be slower because of its division operator, but it does not
require counting the trailing zeros.

unsigned int t = (v | (v - 1)) + 1;
w = t | ((((t & -t) / (v & -v)) >> 1) - 1);

Thanks to Dario Sneidermanis of Argentina, who provided this on November 28, 2009.