Insertion sort

Definition

Let $(X, \le)$ be a totally ordered set. Let $x \in X$ and let $s = (s_1, \dots, s_k)$ be a sorted sequence over $X$ . Let $j$ be the largest index in $\set{0, \dots, k}$ with $j = 0$ or $s_j \le x$ . The insertion of $x$ into $s$ is the sequence $(s_1, \dots, s_j, x, s_{j+1}, \dots, s_k)$ . Insertion sort is the sorting algorithm that, on input $(a_1, \dots, a_n)$ , starts with the empty sequence $s^{(0)}$ . For $i = 1, \dots, n$ it lets $s^{(i)}$ be the insertion of $a_i$ into $s^{(i-1)}$ , and it returns $s^{(n)}$ .

Remarks

In the usual array version, $j$ is found by comparing $x$ with $s_k, s_{k-1}, \dots$ from the right and shifting each larger element one place. The number of shifts equals the number of inversions of the input, the pairs $i < j$ with $a_i > a_j$ . So the running time is $\Theta(n + \mathrm{inv})$ , where $\mathrm{inv}$ is the number of inversions; it is $\Theta(n^2)$ on the reversed list, which has $n(n-1)/2$ inversions. Inserting after all equal elements makes the algorithm stable.

Example: on $(2, 3, 1)$ one gets $s^{(1)} = (2)$ , $s^{(2)} = (2, 3)$ since $2 \le 3$ , and $s^{(3)} = (1, 2, 3)$ , since $j = 0$ for $x = 1$ .

Non-example: appending each $a_i$ at the end of $s^{(i-1)}$ instead of at position $j + 1$ just copies the input, and $(2, 1)$ stays unsorted.

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Sorting algorithm concept
Insertion sort definition

Tag	`5DYWS7DVQJ`
URI	`m2://5DYWS7DVQJ`
Kind	definition
Writing	W0 stub
MSC	68P10, 68W40
Defines	Insertion sort
Uses	Sorting algorithm
Changed	3 November 2026

Insertion sort

Definition

Remarks

Reading path

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