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 worst case. We can reduce it to O(log n) by using binary search.

C/C++

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// 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 sould 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 program to test above function
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;
}

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Java














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// 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]+" "); 


    } 


  


    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  



















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Python














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# 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 xrange(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, 17, 12, 72, 31, 


                        46, 100, 88, 54]) 


  


# Code contributed by Mohit Gupta_OMG  



















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C#














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// 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] + " "); 


    } 


  


    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. 



















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PHP














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<?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 


?> 



















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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. 


        
          
          
          
        

            

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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