# Binary Insertion Sort – Explanation and Implementation

|**Binary Insertion sort** is a variant of Insertion sorting in which proper location to insert the selected element is found using the binary search.

*Read Insertion Sort in detail for complete understanding.*

Binary search reduces the number of comparisons in order to find the correct location in the sorted part of data.

**In worst case scenario** – Normal insertion sort takes O( i ) time in its i^{th }iteration whereas using binary search can reduce it to O(log( i )).

**Note** – Overall time complexity of the algorithm in the worst case is still O(n^{2}) because of the number of swaps required to put every element at the correct location.

#### Implementation of Binary Insertion Sort in C

#include <stdio.h> // A binary search based function to find the position // where item should be inserted 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[] = {4, 10, 3, 1, 9, 15}; int n = sizeof(a)/sizeof(a[0]), i; insertionSort(a, n); printf("Sorted array: "); for (i = 0; i < n; i++) printf("%d ",a[i]); return 0; }

**Output:-**
Sorted array - 1 3 4 9 10 15

#### Implementation of Binary Insertion Sort in Java

In this implementation, I have used library functions for binary search and shifting array to one location right. To understand in detail on how binary search works read this article.

package com.codingeek.algorithms.sorting; import java.util.Arrays; class BinaryInsertionSort{ public static void main(String[] args) { final int[] input = { 4, 10, 3, 1, 9, 15 }; System.out.println("Before Sorting - " + Arrays.toString(input)); new BinaryInsertionSort().sort(input); System.out.println("After Sorting - " + Arrays.toString(input)); } 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; } }

**Output:-**
Before Sorting - [4, 10, 3, 1, 9, 15]
After Sorting - [1, 3, 4, 9, 10, 15]

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