insertion sort

The Problem

See the sorting problem.

The Algorithm

Suppose L={x1,x2,,xn} is the initial list of unsorted elements. The insertion sort algorithmMathworldPlanetmath will construct a new list, containing the elements of L in order, which we will call L. The algorithm constructs this list one element at a time.

Initially L is empty. We then take the first element of L and put it in L. We then take the second element of L and also add it to L, placing it before any elements in L that should come after it. This is done one element at a time until all n elements of L are in L, in sorted order. Thus, each step i consists of looking up the position in L where the element xi should be placed and inserting it there (hence the name of the algorithm). This requires a search, and then the shifting of all the elements in L that come after xi (if L is stored in an array). If storage is in an array, then the binary search algorithm can be used to quickly find xi’s new position in L.

Since at step i, the length of list L is i and the length of list L is n-i, we can implement this algorithm as an in-place sorting algorithm. Each step i results in L[1..i] becoming fully sorted.


This algorithm uses a modified binary search algorithm to find the position in L where an element key should be placed to maintain orderingMathworldPlanetmath.

Algorithm Insertion_Sort(L, n)
Input: A list L of n elements
Output: The list L in sorted order

          for i1 to n do




for ji downto position do




function Binary_Search(L, bottom, top, key)
          if bottomtop then





if key<L[middle] then






In the worst case, each step i requires a shift of i-1 elements for the insertion (consider an input list that is sorted in reverse order). Thus the runtime complexity is 𝒪(n2). Even the optimization of using a binary searchMathworldPlanetmath does not help us here, because the deciding factor in this case is the insertion. It is possible to use a data type with 𝒪(logn) insertion time, giving 𝒪(nlogn) runtime, but then the algorithm can no longer be done as an in-place sorting algorithm. Such data structures are also quite complicated.

A similar algorithm to the insertion sort is the selection sortMathworldPlanetmath, which requires fewer data movements than the insertion sort, but requires more comparisons.

Title insertion sort
Canonical name InsertionSort
Date of creation 2013-03-22 11:44:38
Last modified on 2013-03-22 11:44:38
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 16
Author mathcam (2727)
Entry type Algorithm
Classification msc 68P10
Classification msc 55U40
Classification msc 55U99
Classification msc 55U15
Classification msc 55U10
Classification msc 55P20
Classification msc 55-00
Classification msc 85-00
Classification msc 83-02
Related topic SortingProblem
Related topic BinarySearch
Related topic SelectionSort