How Can We Efficiently Generate All k Combinations of n Items in C ?

Generating k Combinations of n Items Effectively in C

The task of generating all possible combinations of k individuals from a group of n involves employing a clever algorithm that leverages binary representations.

Implementation

The essence of the algorithm lies in creating a list of binary numbers where the first k bits are set to 1, with the remaining n - k bits being set to 0. This binary representation serves as an assignment list, where each bit indicates the inclusion or exclusion of a specific individual in the combination.

Algorithm

The key steps of the algorithm are as follows:

1. Initialize a bitmask with k leading 1's and n - k trailing 0's.
2. Iterate over the bitmask, decoding the binary representation to identify the selected individuals.
3. Generate the combination by printing the selected individuals.
4. Apply the std::prev_permutation function to generate the next combination by permuting the bitmask.

Example

Consider a scenario with n = 5 and k = 3. The algorithm would generate the following combinations:

0 1 2
0 1 3
0 1 4
0 2 3
0 2 4
0 3 4
1 2 3
1 2 4
1 3 4
2 3 4

Analysis

This algorithm effectively exploits binary representations to efficiently generate all possible combinations. The bitmask manipulation provides a straightforward and rapid method for determining the inclusion or exclusion of individuals in the combinations.

Optimization

To further enhance the efficiency, an optimization involves sorting the generated combinations. While not strictly necessary, it ensures a consistent order of combinations for the same input parameters.

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