art-hack's blog

Generating subsets of size K using bits
By art-hack, 4 years ago, In English

Hi Everyone, I encountered this subproblem in a problem.

Suppose we are given an array of integers of size N and want to generate subsets of size K from it. I used this simple backtracking approach to generate the combinations of size K.

Backtracking Code

I was wondering if there is a possible approach to do this by using bits, a simple approach is generating all subsets and processing only the ones that have size K, but in this, we have to generate all 2N subsets.

Bits Solution

Is there a way to do this using bits without generating all the possible subsets?.

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4 years ago, # |
  Vote: I like it +49 Vote: I do not like it

One nice way to do this is to create a vector of $$$n-k$$$ 0's followed by $$$k$$$ 1's. Use std::next_permutation to iterate over all combinations.

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    4 years ago, # ^ |
      Vote: I like it +3 Vote: I do not like it

    Indeed, a really nice solution. I didn't know that next_permutation takes care of duplicates as well, I thought that it will generate duplicates. I will look into how it actually works.

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4 years ago, # |
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You can use Pascal's recursion. Go concatenating the string.

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4 years ago, # |
← Rev. 2   Vote: I like it 0 Vote: I do not like it

I suggest using std::next_permutation or std::prev_permutation for better complexity $$$O(\binom{n}{k})$$$

Sameple code
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    4 years ago, # ^ |
    ← Rev. 2   Vote: I like it 0 Vote: I do not like it

    There is an error in the sample code, in the second line it should be p[N-i-1] = 1;. Instead of generating the first subset, it is creating the last one.

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4 years ago, # |
← Rev. 3   Vote: I like it +6 Vote: I do not like it

Kactl has really short iterative code for this in the "10.5.1 Bit Hacks" section, with the extremely nice properties that the bitsets are generated in increasing order and no other bitsets are generated; here's roughly what it is:

int mask = (1 << k) - 1, r, c;
while(mask <= (1 << n) - (1 << (n-k))) {
    // Code here
    c = mask & -mask, r = mask+c, mask = r | (((r ^ mask) >> 2)/c);
}

The loop bounds I've written only work for $$$n \leq 30$$$ (or 62 for long longs). It's a really good exercise in bit operations to think about why this works.

Note: if you are adapting this code to long longs, make sure you change things like (1 << k) to (1LL << k).

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    4 years ago, # ^ |
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    This is exactly what I was looking for and it looks like that the complexity is even better than the solution that Monogon suggested, as worst-case complexity for next_permutation is O(n). I will now try to think about why it works.

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    4 years ago, # ^ |
    ← Rev. 7   Vote: I like it +18 Vote: I do not like it

    In case you need to find previous subset instead of next subset, you can you a trick to get the inverse function: prev = ~next(~mask), next = ~prev(~mask)

    Generating next subset
    Generating prev subset

    Moreover, if you want to get the kth previous or kth next subset, you can use lexicographical dynamic programming for it.

    Here I use mask of long long, you can change to bitset<> to store larger but it will be a bit complicated

    Simple Implementation
    Full implementation (with detail comment)
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      4 years ago, # ^ |
      ← Rev. 2   Vote: I like it 0 Vote: I do not like it

      In case guys interesting, there are some other effecient or non-effecient way to find it. (Sorry ! here I only provides implementations and not describe how these function works perfectly)

      Implementation 1 - O(n log n)
      Implementation 2
      Implementation 3
      Implementation 4
      Implementation 5
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      23 months ago, # ^ |
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      Can you please explain the divide part (x ^ higher) / right) >> 2

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        21 month(s) ago, # ^ |
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        You can output step-to-step bitwise for a few examples to understand it better. As it just does not naturally occur but I compact the code whenever it is possible.

        But my ideas are:

        • I must not change the number of set bits unless this is the last iteration, how can I detect it and what should it return in that case?
        • Some bits must stay the same, and only a few are shifted, how do I detect which and how can I move them correctly
        • Can I make it better without using for, loop, workcase, constants, precomputed table?
        • Can I make it better without using many multiplication, division, and modulo operations ?
        • Can I make it shorter and faster with fewer bit manipulations and less variables defined?

        IMO, you can come up to yourself a better or/and simpler solution and I'd love to see that.

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