# C Program for Binary Insertion Sort

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

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

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