How does the std::next_permutation algorithm work and what are its key components?

std::next_permutation Implementation Explained

Question:
Can you explain the functionality, variable roles, and correctness of the std::next_permutation algorithm?

Answer:

How It Works:

std::next_permutation rearranges a given sequence of elements into the next lexicographically greater permutation. It does so by locating the first element i where i j for some j after it and then finding the next larger element k such that i k from the end of the sequence.

1. Find the first decreasing element (i):

   • Iterate through the sequence from right to left until an element i is found where i j for the next element j.

2. Find the next larger element (k):

   • Iterate from the right end of the sequence until an element k is found where i k.

3. Swap i and k:

   • Exchange the positions of elements i and k to introduce a higher value at the decreasing point.

4. Reverse the subsequence after j:

   • Since swapping i and k may have disrupted the descending order of the remaining elements after i, they are sorted in ascending order by reversing that subsequence.

Variable Roles:

• i: Pointer to the first decreasing element.
• j: Pointer to the next element after i.
• k: Pointer to the next larger element from the end.

Correctness Sketch:

The algorithm preserves the property that the subsequence from i to the end remains in descending order throughout the process.

1. Descending Order after Swapping:

   • The initial descending order from i to the end is maintained because the elements after j are not modified in the swap.

2. Lexicographically Smaller

   • Swapping i and k ensures that the resulting permutation is lexicographically larger than the previous one, as k is the next larger element encountered.

3. Permutation Exhaustion:

   • When there is no decreasing element i, the entire sequence is in descending order, indicating the last permutation has been reached.

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