How does std::next_permutation generate the next lexicographically greater permutation?

Understanding std::next_permutation

std::next_permutation is an algorithm that computes the next lexicographically greater permutation of a container of elements. This means that it changes the elements in the container in such a way that the overall ordering remains consistent, while maximizing the elements in some defined order.

How Does it Work?

Key Insight: The algorithm assumes that elements can be treated as digits, and permutations as numbers. Ordering permutations, then, becomes equivalent to ordering numbers in ascending order.

Implementation:

1. Main Loop: The algorithm enters a loop that checks if the elements to the right of the current element i are in descending order.

   • If they are, it means there are no more permutations of the right-hand elements.
   • If they are not, it proceeds with the permutation process.
2. Finding the Leftmost Ascendable Element: The algorithm decrements i until it finds an element j where i < j. This indicates that i and j are part of a subsequence that is not in descending order.
3. Finding the Next Largest Element: It finds the next largest element k from the end of the container such that i < k.
4. Swapping and Reversing: The algorithm swaps i and k, essentially moving the next largest element to the left of the current element. It then reverses the sequence from j to the end, ensuring that the right-hand elements remain in ascending order.

Meaning of Variables:

• i: The current element under consideration.
• j: The element before i (used for descending order check).
• k: The next largest element from the end of the sequence.

Proof of Correctness (Sketch)

• Monotonicity: It proves that if the sequence before the loop was lexicographically ordered, the sequence after the loop will also be lexicographically ordered.
• Termination: It shows that the algorithm will eventually reach a point where i can no longer be decremented, indicating that the last permutation has been computed.

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