Binary Insertion Sort

Binary insertion sort is a sorting algorithm which is similar to the insertion sort, but instead of using linear search to find the location where an element should be inserted, we use binary search. Thus, we reduce the comparative value of inserting a single element from O (N) to O (log N).

It is a flexible algorithm, which means it works faster when the same given members are already heavily sorted, i.e., the current location of the feature is closer to its actual location in the sorted list.

It is a stable filtering algorithm – elements with the same values ??appear in the same sequence in the last order as they were in the first list.

Applications of Binary Insertion sort:

• Binary insertion sort works best when the array has a lower number of items.
• When doing quick sort or merge sort, when the subarray size becomes smaller (say <= 25 elements), it is best to use a binary insertion sort.
• This algorithm also works when the cost of comparisons between keys is high enough. For example, if we want to filter multiple strings, the comparison performance of two strings will be higher.

How Does Binary insertion sort Work?

• In the binary insertion sort mode, we divide the same members into two subarrays – filtered and unfiltered. The first element of the same members is in the organized subarray, and all other elements are unplanned.
• Then we iterate from the second element to the last. In the repetition of the i-th, we make the current object our “key”. This key is a feature that we should add to our existing list below.
• In order to do this, we first use a binary search on the sorted subarray below to find the location of an element larger than our key. Let’s call this position “pos.” We then right shift all the elements from pos to 1 and created Array[pos] = key.
• We can note that in every i-th multiplication, the left part of the array till (i – 1) is already sorted.

Approach to implement Binary Insertion sort:

• Iterate the array from the second element to the last element.
• Store the current element A[i] in a variable key.
• Find the position of the element just greater than A[i] in the subarray from A[0] to A[i-1] using binary search. Say this element is at index pos.
• Shift all the elements from index pos to i-1 towards the right.
• A[pos] = key.

Below is the implementation for the above approach:

C++

// C program for implementation of
// binary insertion sort
#include <iostream>
using namespace std;

// A binary search based function
// to find the position
// where item should be inserted
// in a[low..high]
int binarySearch(int a[], int item,
                int low, int high)
{
    if (high <= low)
        return (item > a[low]) ?
                (low + 1) : low;

    int mid = (low + high) / 2;

    if (item == a[mid])
        return mid + 1;

    if (item > a[mid])
        return binarySearch(a, item,
                            mid + 1, high);
    return binarySearch(a, item, low,
                        mid - 1);
}

// Function to sort an array a[] of size 'n'
void insertionSort(int a[], int n)
{
    int i, loc, j, k, selected;

    for (i = 1; i < n; ++i)
    {
        j = i - 1;
        selected = a[i];

        // find location where selected should be inserted
        loc = binarySearch(a, selected, 0, j);

        // Move all elements after location to create space
        while (j >= loc)
        {
            a[j + 1] = a[j];
            j--;
        }
        a[j + 1] = selected;
    }
}

// Driver Code
int main()
{
    int a[]
        = { 37, 23, 0, 17, 12, 72, 31, 46, 100, 88, 54 };
    int n = sizeof(a) / sizeof(a[0]), i;

    insertionSort(a, n);

    cout <<"Sorted array: \n";
    for (i = 0; i < n; i++)
        cout <<" "<< a[i];

    return 0;
}

// this code is contribution by shivanisinghss2110

C

// C program for implementation of
// binary insertion sort
#include <stdio.h>

// A binary search based function
// to find the position
// where item should be inserted
// in a[low..high]
int binarySearch(int a[], int item,
                 int low, int high)
{
    if (high <= low)
        return (item > a[low]) ?
                (low + 1) : low;

    int mid = (low + high) / 2;

    if (item == a[mid])
        return mid + 1;

    if (item > a[mid])
        return binarySearch(a, item,
                            mid + 1, high);
    return binarySearch(a, item, low,
                        mid - 1);
}

// Function to sort an array a[] of size 'n'
void insertionSort(int a[], int n)
{
    int i, loc, j, k, selected;

    for (i = 1; i < n; ++i)
    {
        j = i - 1;
        selected = a[i];

        // find location where selected should be inserted
        loc = binarySearch(a, selected, 0, j);

        // Move all elements after location to create space
        while (j >= loc)
        {
            a[j + 1] = a[j];
            j--;
        }
        a[j + 1] = selected;
    }
}

// Driver Code
int main()
{
    int a[]
        = { 37, 23, 0, 17, 12, 72, 31, 46, 100, 88, 54 };
    int n = sizeof(a) / sizeof(a[0]), i;

    insertionSort(a, n);

    printf("Sorted array: \n");
    for (i = 0; i < n; i++)
        printf("%d ", a[i]);

    return 0;
}

Java

// Java Program implementing
// binary insertion sort

import java.util.Arrays;
class GFG
{

    public static void main(String[] args)
    {
        final int[] arr = { 37, 23, 0,   17, 12, 72,
                            31, 46, 100, 88, 54 };

        new GFG().sort(arr);

        for (int i = 0; i < arr.length; i++)
            System.out.print(arr[i] + " ");
    }

    // Driver Code
    public void sort(int array[])
    {
        for (int i = 1; i < array.length; i++)
        {
            int x = array[i];

            // Find location to insert
            // using binary search
            int j = Math.abs(
                Arrays.binarySearch(array, 0,
                                    i, x) + 1);

            // Shifting array to one
            // location right
            System.arraycopy(array, j,
                             array, j + 1, i - j);

            // Placing element at its
            // correct location
            array[j] = x;
        }
    }
}

// Code contributed by Mohit Gupta_OMG

Python3

# Python Program implementation
# of binary insertion sort


def binary_search(arr, val, start, end):

    # we need to distinguish whether we
    # should insert before or after the
    # left boundary. imagine [0] is the last
    # step of the binary search and we need
    # to decide where to insert -1
    if start == end:
        if arr[start] > val:
            return start
        else:
            return start+1

    # this occurs if we are moving
    # beyond left's boundary meaning
    # the left boundary is the least
    # position to find a number greater than val
    if start > end:
        return start

    mid = (start+end)//2
    if arr[mid] < val:
        return binary_search(arr, val, mid+1, end)
    elif arr[mid] > val:
        return binary_search(arr, val, start, mid-1)
    else:
        return mid


def insertion_sort(arr):
    for i in range(1, len(arr)):
        val = arr[i]
        j = binary_search(arr, val, 0, i-1)
        arr = arr[:j] + [val] + arr[j:i] + arr[i+1:]
    return arr


print("Sorted array:")
print(insertion_sort([37, 23, 0, 31, 22, 17, 12, 72, 31, 46, 100, 88, 54]))

# Code contributed by Mohit Gupta_OMG

C#

// C# Program implementing
// binary insertion sort
using System;

class GFG {

    public static void Main()
    {
        int[] arr = { 37, 23, 0,   17, 12, 72,
                      31, 46, 100, 88, 54 };

        sort(arr);

        for (int i = 0; i < arr.Length; i++)
            Console.Write(arr[i] + " ");
    }

    // Driver Code
    public static void sort(int[] array)
    {
        for (int i = 1; i < array.Length; i++)
        {
            int x = array[i];

            // Find location to insert using
            // binary search
            int j = Math.Abs(
                Array.BinarySearch(array,
                                   0, i, x) + 1);

            // Shifting array to one location right
            System.Array.Copy(array, j,
                              array, j + 1,
                              i - j);

            // Placing element at its correct
            // location
            array[j] = x;
        }
    }
}

// This code is contributed by nitin mittal.

PHP

<?php
// PHP program for implementation of
// binary insertion sort

// A binary search based function to find
// the position where item should be
// inserted in a[low..high]
function binarySearch($a, $item, $low, $high)
{

    if ($high <= $low)
        return ($item > $a[$low]) ?
                       ($low + 1) : $low;

    $mid = (int)(($low + $high) / 2);

    if($item == $a[$mid])
        return $mid + 1;

    if($item > $a[$mid])
        return binarySearch($a, $item,
                            $mid + 1, $high);

    return binarySearch($a, $item, $low,
                            $mid - 1);
}

// Function to sort an array a of size 'n'
function insertionSort(&$a, $n)
{
    $i; $loc; $j; $k; $selected;

    for ($i = 1; $i < $n; ++$i)
    {
        $j = $i - 1;
        $selected = $a[$i];

        // find location where selected
        // item should be inserted
        $loc = binarySearch($a, $selected, 0, $j);

        // Move all elements after location
        // to create space
        while ($j >= $loc)
        {
            $a[$j + 1] = $a[$j];
            $j--;
        }
        $a[$j + 1] = $selected;
    }
}

// Driver Code
$a = array(37, 23, 0, 17, 12, 72,
           31, 46, 100, 88, 54);

$n = sizeof($a);

insertionSort($a, $n);

echo "Sorted array:\n";
for ($i = 0; $i < $n; $i++)
    echo "$a[$i] ";

// This code is contributed by
// Adesh Singh
?>

Javascript

<script>
// Javascript Program implementing
// binary insertion sort

function binarySearch(a, item, low, high)
{

    if (high <= low)
        return (item > a[low]) ?
                       (low + 1) : low;

    mid = Math.floor((low + high) / 2);

    if(item == a[mid])
        return mid + 1;

    if(item > a[mid])
        return binarySearch(a, item,
                            mid + 1, high);

    return binarySearch(a, item, low,
                            mid - 1);
}

function sort(array)
{
    for (let i = 1; i < array.length; i++)
        {
            let j = i - 1;
            let x = array[i];

            // Find location to insert
            // using binary search
            let loc = Math.abs(
                binarySearch(array, x,
                                    0, j));

            // Shifting array to one
            // location right

            while (j >= loc)
            {
                array[j + 1] = array[j];
                j--;
            }

            // Placing element at its
            // correct location
            array[j+1] = x;
        }
}

// Driver Code
let arr=[ 37, 23, 0,   17, 12, 72,
                            31, 46, 100, 88, 54];
sort(arr);

for (let i = 0; i < arr.length; i++)
    document.write(arr[i] + " ");


// This code is contributed by unknown2108
</script>
// C program for implementation of
// binary insertion sort
#include <stdio.h>

// A binary search based function
// to find the position
// where item should be inserted
// in a[low..high]
int binarySearch(int a[], int item,
                int low, int high)
{
    if (high <= low)
        return (item > a[low]) ?
                (low + 1) : low;

    int mid = (low + high) / 2;

    if (item == a[mid])
        return mid + 1;

    if (item > a[mid])
        return binarySearch(a, item,
                            mid + 1, high);
    return binarySearch(a, item, low,
                        mid - 1);
}

// Function to sort an array a[] of size 'n'
void insertionSort(int a[], int n)
{
    int i, loc, j, k, selected;

    for (i = 1; i < n; ++i)
    {
        j = i - 1;
        selected = a[i];

        // find location where selected should be inserted
        loc = binarySearch(a, selected, 0, j);

        // Move all elements after location to create space
        while (j >= loc)
        {
            a[j + 1] = a[j];
            j--;
        }
        a[j + 1] = selected;
    }
}

// Driver Code
int main()
{
    int a[]
        = { 37, 23, 0, 17, 12, 72, 31, 46, 100, 88, 54 };
    int n = sizeof(a) / sizeof(a[0]), i;

    insertionSort(a, n);

    printf("Sorted array: \n");
    for (i = 0; i < n; i++)
        printf("%d ", a[i]);

    r// C program for implementation of
// binary insertion sort
#include <stdio.h>

// A binary search based function
// to find the position
// where item should be inserted
// in a[low..high]
int binarySearch(int a[], int item,
                int low, int high)
{
    if (high <= low)
        return (item > a[low]) ?
                (low + 1) : low;

    int mid = (low + high) / 2;

    if (item == a[mid])
        return mid + 1;

    if (item > a[mid])
        return binarySearch(a, item,
                            mid + 1, high);
    return binarySearch(a, item, low,
                        mid - 1);
}

// Function to sort an array a[] of size 'n'
void insertionSort(int a[], int n)
{
    int i, loc, j, k, selected;

    for (i = 1; i < n; ++i)
    {
        j = i - 1;
        selected = a[i];

        // find location where selected should be inseretd
        loc = binarySearch(a, selected, 0, j);

        // Move all elements after location to create space
        while (j >= loc)
        {
            a[j + 1] = a[j];
            j--;
        }
        a[j + 1] = selected;
    }
}

// Driver Code
int main()
{
    int a[]
        = { 37, 23, 0, 17, 12, 72, 31, 46, 100, 88, 54 };
    int n = sizeof(a) / sizeof(a[0]), i;

    insertionSort(a, n);

    printf("Sorted array: \n");
    for (i = 0; i < n; i++)
        printf("%d ", a[i]);

    // C program for implementation of
// binary insertion sort
#include <stdio.h>

// A binary search based function
// to find the position
// where item should be inserted
// in a[low..high]
int binarySearch(int a[], int item,
                int low, int high)
{
    if (high <= low)
        return (item > a[low]) ?
                (low + 1) : low;

    int mid = (low + high) / 2;

    if (item == a[mid])
        return mid + 1;

    if (item > a[mid])
        return binarySearch(a, item,
                            mid + 1, high);
    return binarySearch(a, item, low,
                        mid - 1);
}

// Function to sort an array a[] of size 'n'
void insertionSort(int a[], int n)
{
    int i, loc, j, k, selected;

    for (i = 1; i < n; ++i)
    {
        j = i - 1;
        selected = a[i];

        // find location where selected should be inseretd
        loc = binarySearch(a, selected, 0, j);

        // Move all elements after location to create space
        while (j >= loc)
        {
            a[j + 1] = a[j];
            j--;
        }
        a[j + 1] = selected;
    }
}

// Driver Code
int main()
{
    int a[]
        = { 37, 23, 0, 17, 12, 72, 31, 46, 100, 88, 54 };
    int n = sizeof(a) / sizeof(a[0]), i;

    insertionSort(a, n);

    printf("Sorted array: \n");
    for (i = 0; i < n; i++)
        printf("%d ", a[i]);

// C program for implementation of
// binary insertion sort
#include <stdio.h>

// A binary search based function
// to find the position
// where item should be inserted
// in a[low..high]
int binarySearch(int a[], int item,
                int low, int high)
{
    if (high <= low)
        return (item > a[low]) ?
                (low + 1) : low;

    int mid = (low + high) / 2;

    if (item == a[mid])
        return mid + 1;

    if (item > a[mid])
        return binarySearch(a, item,
                            mid + 1, high);
    return binarySearch(a, item, low,
                        mid - 1);
}

// Function to sort an array a[] of size 'n'
void insertionSort(int a[], int n)
{
    int i, loc, j, k, selected;

    for (i = 1; i < n; ++i)
    {
        j = i - 1;
        selected = a[i];

        // find location where selected should be inserted
        loc = binarySearch(a, selected, 0, j);

        // Move all elements after location to create space
        while (j >= loc)
        {
            a[j + 1] = a[j];
            j--;
        }
        a[j + 1] = selected;
    }
}

// Driver Code
int main()
{
    int a[]
        = { 37, 23, 0, 17, 12, 72, 31, 46, 100, 88, 54 };
    int n = sizeof(a) / sizeof(a[0]), i;

    insertionSort(a, n);

    printf("Sorted array: \n");
    for (i = 0; i < n; i++)
        printf("%d ", a[i]);
// C program for implementation of
// binary insertion sort
#include <stdio.h>

// A binary search based function
// to find the position
// where item should be inserted
// in a[low..high]
int binarySearch(int a[], int item,
                int low, int high)
{
    if (high <= low)
        return (item > a[low]) ?
                (low + 1) : low;

    int mid = (low + high) / 2;

    if (item == a[mid])
        return mid + 1;

    if (item > a[mid])
        return binarySearch(a, item,
                            mid + 1, high);
    return binarySearch(a, item, low,
                        mid - 1);
}

// Function to sort an array a[] of size 'n'
void insertionSort(int a[], int n)
{
    int i, loc, j, k, selected;

    for (i = 1; i < n; ++i)
    {
        j = i - 1;
        selected = a[i];

        // find location where selected should be inseretd
        loc = binarySearch(a, selected, 0, j);

        // Move all elements after location to create space
        while (j >= loc)
        {
            a[j + 1] = a[j];
            j--;
        }
        a[j + 1] = selected;
    }
}

// Driver Code
int main()
{
    int a[]
        = { 37, 23, 0, 17, 12, 72, 31, 46, 100, 88, 54 };
    int n = sizeof(a) / sizeof(a[0]), i;

    insertionSort(a, n);

    printf("Sorted array: \n");
    for (i = 0; i < n; i++)
        printf("%d ", a[i]);// C program for implementation of
// binary insertion sort
#include <stdio.h>

// A binary search based function
// to find the position
// where item should be inserted
// in a[low..high]
int binarySearch(int a[], int item,
                int low, int high)
{
    if (high <= low)
        return (item > a[low]) ?
                (low + 1) : low;

    int mid = (low + high) / 2;

    if (item == a[mid])
        return mid + 1;

    if (item > a[mid])
        return binarySearch(a, item,
                            mid + 1, high);
    return binarySearch(a, item, low,
                        mid - 1);
}

// Function to sort an array a[] of size 'n'
void insertionSort(int a[], int n)
{
    int i, loc, j, k, selected;

    for (i = 1; i < n; ++i)
    {
        j = i - 1;
        selected = a[i];

        // find location where selected should be inseretd
        loc = binarySearch(a, selected, 0, j);

        // Move all elements after location to create space
        while (j >= loc)
        {
            a[j + 1] = a[j];
            j--;
        }
        a[j + 1] = selected;
    }
}

// Driver Code
int main()
{
    int a[]
        = { 37, 23, 0, 17, 12, 72, 31, 46, 100, 88, 54 };
    int n = sizeof(a) / sizeof(a[0]), i;

    insertionSort(a, n);

    printf("Sorted array: \n");
    for (i = 0; i < n; i++)
        printf("%d ", a[i])

Output

Sorted array: 
 0 12 17 23 31 37 46 54 72 88 100

Time Complexity: The algorithm as a whole still has a running worst-case running time of O(n²) because of the series of swaps required for each insertion. 

Another approach: Following is an iterative implementation of the above recursive code

C++

#include <iostream>
using namespace std;

// iterative implementation
int binarySearch(int a[], int item, int low, int high)
{
    while (low <= high) {
        int mid = low + (high - low) / 2;
        if (item == a[mid])
            return mid + 1;
        else if (item > a[mid])
            low = mid + 1;
        else
            high = mid - 1;
    }

    return low;
}

// Function to sort an array a[] of size 'n'
void insertionSort(int a[], int n)
{
    int i, loc, j, k, selected;

    for (i = 1; i < n; ++i) {
        j = i - 1;
        selected = a[i];

        // find location where selected should be inserted
        loc = binarySearch(a, selected, 0, j);

        // Move all elements after location to create space
        while (j >= loc) {
            a[j + 1] = a[j];
            j--;
        }
        a[j + 1] = selected;
    }
}

// Driver Code
int main()
{
    int a[]
        = { 37, 23, 0, 17, 12, 72, 31, 46, 100, 88, 54 };
    int n = sizeof(a) / sizeof(a[0]), i;

    insertionSort(a, n);

    cout <<"Sorted array: \n";
    for (i = 0; i < n; i++)
        cout <<" "<< a[i];

    return 0;
}

// This code is contributed by shivanisinghss2110.

C

#include <stdio.h>

// iterative implementation
int binarySearch(int a[], int item, int low, int high)
{
    while (low <= high) {
        int mid = low + (high - low) / 2;
        if (item == a[mid])
            return mid + 1;
        else if (item > a[mid])
            low = mid + 1;
        else
            high = mid - 1;
    }

    return low;
}

// Function to sort an array a[] of size 'n'
void insertionSort(int a[], int n)
{
    int i, loc, j, k, selected;

    for (i = 1; i < n; ++i) {
        j = i - 1;
        selected = a[i];

        // find location where selected should be inserted
        loc = binarySearch(a, selected, 0, j);

        // Move all elements after location to create space
        while (j >= loc) {
            a[j + 1] = a[j];
            j--;
        }
        a[j + 1] = selected;
    }
}

// Driver Code
int main()
{
    int a[]
        = { 37, 23, 0, 17, 12, 72, 31, 46, 100, 88, 54 };
    int n = sizeof(a) / sizeof(a[0]), i;

    insertionSort(a, n);

    printf("Sorted array: \n");
    for (i = 0; i < n; i++)
        printf("%d ", a[i]);

    return 0;
}
// contributed by tmeid

Java

import java.io.*;

class GFG {

// iterative implementation
static int binarySearch(int a[], int item, int low, int high)
{
    while (low <= high) {
        int mid = low + (high - low) / 2;
        if (item == a[mid])
            return mid + 1;
        else if (item > a[mid])
            low = mid + 1;
        else
            high = mid - 1;
    }

    return low;
}

// Function to sort an array a[] of size 'n'
static void insertionSort(int a[], int n)
{
    int i, loc, j, k, selected;

    for (i = 1; i < n; ++i) {
        j = i - 1;
        selected = a[i];

        // find location where selected should be inserted
        loc = binarySearch(a, selected, 0, j);

        // Move all elements after location to create space
        while (j >= loc) {
            a[j + 1] = a[j];
            j--;
        }
        a[j + 1] = selected;
    }
}

// Driver Code
public static void main (String[] args)
{
    int a[]
        = { 37, 23, 0, 17, 12, 72, 31, 46, 100, 88, 54 };
    int n = a.length, i;

    insertionSort(a, n);

    System.out.println("Sorted array:");
    for (i = 0; i < n; i++)
        System.out.print(a[i] +" ");

}
}

// This code is contributed by shivanisinghss2110.

Python3

# iterative implementation
def binarySearch(a, item, low, high):
    while (low <= high):
        mid = low + (high - low) // 2
        if (item == a[mid]):
            return mid + 1
        elif (item > a[mid]):
            low = mid + 1
        else:
            high = mid - 1
    return low

# Function to sort an array a[] of size 'n'
def insertionSort(a, n):
    for i in range (n):
        j = i - 1
        selected = a[i]

        # find location where selected should be inserted
        loc = binarySearch(a, selected, 0, j)

        # Move all elements after location to create space
        while (j >= loc):
            a[j + 1] = a[j]
            j-=1
        a[j + 1] = selected

# Driver Code
a = [37, 23, 0, 17, 12, 72, 31, 46, 100, 88, 54]
n = len(a)
insertionSort(a, n)
print("Sorted array: ")
for i in range (n):
    print(a[i], end=" ")

# This code is contributed by shivanisinghss2110

C#

using System;

class GFG {

// iterative implementation
static int binarySearch(int []a, int item, int low, int high)
{
    while (low <= high) {
        int mid = low + (high - low) / 2;
        if (item == a[mid])
            return mid + 1;
        else if (item > a[mid])
            low = mid + 1;
        else
            high = mid - 1;
    }

    return low;
}

// Function to sort an array a[] of size 'n'
static void insertionSort(int []a, int n)
{
    int i, loc, j, selected;

    for (i = 1; i < n; ++i) {
        j = i - 1;
        selected = a[i];

        // find location where selected should be inserted
        loc = binarySearch(a, selected, 0, j);

        // Move all elements after location to create space
        while (j >= loc) {
            a[j + 1] = a[j];
            j--;
        }
        a[j + 1] = selected;
    }
}

// Driver Code
public static void Main (String[] args)
{
    int []a = { 37, 23, 0, 17, 12, 72, 31, 46, 100, 88, 54 };
    int n = a.Length, i;

    insertionSort(a, n);

    Console.WriteLine("Sorted array:");
    for (i = 0; i < n; i++)
        Console.Write(a[i] +" ");

}
}

// This code is contributed by shivanisinghss2110

Javascript

<script>
// iterative implementation
function binarySearch( a,  item,  low,  high)
{
    while (low <= high) {
        var mid = low + (high - low) / 2;
        if (item == a[mid])
            return mid + 1;
        else if (item > a[mid])
            low = mid + 1;
        else
            high = mid - 1;
    }

    return low;
}

// Function to sort an array a[] of size 'n'
function insertionSort(a, n)
{
    var i, loc, j, k, selected;

    for (i = 1; i < n; ++i) {
        j = i - 1;
        selected = a[i];

        // find location where selected should be inserted
        loc = binarySearch(a, selected, 0, j);

        // Move all elements after location to create space
        while (j >= loc) {
            a[j + 1] = a[j];
            j--;
        }
        a[j + 1] = selected;
    }
}

// Driver Code
    var a = [ 37, 23, 0, 17, 12, 72, 31, 46, 100, 88, 54 ];
    var n = a.length, i;

    insertionSort(a, n);

    document.write("Sorted array:" + "<br>");
    for (i = 0; i < n; i++)
        document.write(a[i] +" ");


// This code is contributed by shivanisinghss2110
</script>

Output

Sorted array: 
 0 12 17 23 31 37 46 54 72 88 100

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