Binary Insertion Sort

We can use binary search to reduce the number of comparisons in normal insertion sort. Binary Insertion Sort uses binary search to find the proper location to insert the selected item at each iteration. 
In normal insertion sort, it takes O(n) comparisons (at nth iteration) in the worst case. We can reduce it to O(log n) by using binary search.




// 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 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);
 
    cout <<"Sorted array: \n";
    for (i = 0; i < n; i++)
        cout <<" "<< a[i];
 
    return 0;
}
 
// this code is contribution by shivanisinghss2110




// 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]);
 
    return 0;
}




// 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




# Python Program implementation
# of binary insertion sort
 
 
def binary_search(arr, val, start, end):
     
    # we need to distinugish 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# 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 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
?>




<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 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]);
 
    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 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 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




#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 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);
 
    cout <<"Sorted array: \n";
    for (i = 0; i < n; i++)
        cout <<" "<< a[i];
 
    return 0;
}
 
// This code is contributed by shivanisinghss2110.




#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 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]);
 
    return 0;
}
// contributed by tmeid




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 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
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.




# 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 inseretd
        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




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 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
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




<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 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
    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 

This article is contributed by Amit Auddy. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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