Insertion Sort

Insertion Sort is an algorithm that sorts an array of $n$ numbers $a_0,a_1,\dots ,a_{n-1}$ in non-decreasing order in $\mathcal{O}(n^2)$ time and $\mathcal{O}(1)$ space.

Tip

This problem can also be solved by Selection Sort in $\mathcal{O}(n^2)$ time and $\mathcal{O}(1)$ space.

Tip

This problem can also be solved by Bubble Sort in $\mathcal{O}(n^2)$ time and $\mathcal{O}(1)$ space.

Tip

This problem can also be solved by Merge Sort in $\mathcal{O}(n\log n)$ time and $\mathcal{O}(n)$ space.

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This problem can also be solved by Heapsort in $\mathcal{O}(n\log n)$ time and $\mathcal{O}(1)$ space.

Tip

This problem can also be solved by Quicksort in $\mathcal{O}(n\log n)$ time and $\mathcal{O}(\log n)$ space.

Tip

This problem can also be solved by Quickselect in $\mathcal{O}(n^2)$ time and $\mathcal{O}(n)$ space.

Algorithm 0

0. Sort $a_0,a_1,\dots ,a_{n-2}$ recursively.
1. Insert $a_{n-1}$ into an appropriate position in $a_0,a_1,\dots ,a_{n-2}$.

This algorithm solves the problem in $\mathcal{O}(n^2)$ time and $\mathcal{O}(1)$ space.

void insertion_sort(int n, std::vector<int> &a) {
	for (int i = 1; i < n; i++) {
		int x = a[i], j = i;
		for (; j > 0 && a[j - 1] > x; j--) {
			a[j] = a[j - 1];
		}
		a[j] = x;
	}
}

Algorithm 1

Based on Algorithm 0, applying Binary Search to find the appropriate position yields an algorithm that solves the problem in $\mathcal{O}(n^2)$ time and $\mathcal{O}(1)$ space.

void insertion_sort(int n, std::vector<int> &a) {
	for (int i = 1; i < n; i++) {
		std::rotate(std::ranges::lower_bound(a | std::views::take(i), a[i]), a.begin() + i, a.begin() + i + 1);
	}
}