Computing Combinations on basis of number of bits set - c++

Question

I need to compute all possible combinations of n things selected r at a time where 0<=r<=n, One method to do so is generating the numbers up to 0 to 2^n-1. But I need to generate these numbers such that the numbers should be sorted on the basis of the number of bits set in that number. for n=3:
0 // numbers with 0 bits set
1 2 4 // numbers with 1 bits set
3 5 6 // numbers with 2 bits set
7 // numbers with 3 bits set
I need to know how to generate the numbers such that they are sorted in increasing/decreasing order of bits set?

Answer 1

Implement regular algorithm to generate the combinations, but also hold an additional array where you store the numbers sorted accoding to the 1-bits set. Then for each combination generated replace the numbers with the numbers sitting in the corresponding position minus one in the array sorted as I described.

Answer 2

Iterating over all combinations of some some number of items is covered nicely by quant_dev here.

Answer 3

Here is a simple way function that counts the number of bits set in a number's representation:
// Counts how many bits are set in the representation of the input number n
int numOfBitsSet(int n)
{
int cnt = 0;
while (n != 0)
{
cnt += (n & 1);
n = n >> 1;
}
return cnt;
}
And here is how you could use it in a (C++11) program that does what you want:
#include <algorithm>
#include <vector>
#include <iostream>
#include <iterator>
using namespace std;
int main()
{
// For instance...
int n = 3;
// Fill up a vector of 2^n entries (0 .. 2^(n - 1))
vector<int> v(1 << n);
iota(begin(v), end(v), 0);
// For each number of bits...
for (size_t i = 0; i <= n; i++)
{
cout << "Numbers with " << i << " bits set: ";
// Find the first number with i bits set...
auto it = find_if(begin(v), end(v), [i] (int x) {
return (numOfBitsSet(x) == i);
});
while (it != end(v))
{
cout << *it << " ";
// Find the next number with i bits set...
it = find_if(next(it), end(v), [i] (int x) {
return (numOfBitsSet(x) == i);
});
}
cout << endl;
}
}
If C++11 is not an option for you, you will have to use functors instead of lambdas, and replace std::iota with a manual loop:
#include <algorithm>
#include <vector>
#include <iostream>
#include <iterator>
using namespace std;
struct bit_count_filter
{
bit_count_filter(int i) : _i(i) { }
bool operator () (int x) const { return numOfBitsSet(x) == _i; }
int _i;
};
int main()
{
// For instance...
int n = 3;
// Fill up a vector of 2^n entries (0 .. 2^(n - 1))
vector<int> v(1 << n);
for (size_t i = 0; i < v.size(); i++)
{
v[i] = i;
}
// For each number of bits...
for (size_t i = 0; i <= n; i++)
{
cout << "Numbers with " << i << " bits set: ";
// Find the first number with i bits set...
auto it = find_if(begin(v), end(v), bit_count_filter(i));
while (it != end(v))
{
cout << *it << " ";
// Find the next number with i bits set...
it = find_if(next(it), end(v), bit_count_filter(i));
}
cout << endl;
}
}

Answer 4

You could do it recursively:
void setnbits(unsigned int cur, int n, int toset, int max)
{
if(toset == 0)
{
printf("%d ", cur >> (n + 32 - max) , n);
return;
}
toset--;
for(int i = 1 ; i <= n-toset ; i++)
{
setnbits((cur >> i) | 0x80000000, n-i, toset , max);
}
}
Could be called like:
for(int z = 0 ; z < 4 ; z++)
{
printf("%d bits: ", z);
setnbits(0, 3, z, 3);
printf("\n");
}
prints:
0 bits: 0
1 bits: 1 2 4
2 bits: 3 5 6
3 bits: 7
The numbers are not guaranteed to be in numerical order.

Answer 5

That's pretty easy.
There are two cases:
1) Last 1-bit has 0-bit before:
000111001001 -> 000111001010.
You should just move it to the left
2) There is a chain of 1-bits:
000110111100 -> 000111000111
Then you should move last 1-bit to the nearest 0-bit on the left(before the chain), and move all another bits of that chain to the right.
You'll get this way all needed numbers in increasing order.