How does the binary insertion sort algorithm sort elements?

The binary insertion sort algorithm sorts elements by inserting them into their correct position within a sorted array using binary search.

Binary insertion sort is a variant of the simple insertion sort algorithm. The primary difference is that while insertion sort scans all elements of the sorted part to find the correct position of the element, binary insertion sort uses binary search. This reduces the number of comparisons in the worst case scenario, making it more efficient for larger lists.

The algorithm starts by assuming the first element of the array is a sorted part. It then takes the next element and uses binary search to find the correct position in the sorted part of the array. Binary search works by repeatedly dividing the sorted part in half until the correct position is found. If the element is larger than the middle element of the sorted part, the search continues on the right half. If it's smaller, the search continues on the left half. This process is repeated until the correct position is found.

Once the position is found, the algorithm makes space for the new element by shifting all larger elements to the right. The new element is then inserted in the correct position. This process is repeated for all elements in the array.

It's important to note that while binary insertion sort reduces the number of comparisons, it does not reduce the number of swaps. Each insertion still requires shifting all larger elements, which takes linear time. Therefore, the time complexity of binary insertion sort is still quadratic in the worst case scenario. However, in practice, binary insertion sort can be faster than simple insertion sort for larger lists due to the reduced number of comparisons.

In summary, binary insertion sort is a variant of insertion sort that uses binary search to find the correct position of elements. This makes it more efficient for larger lists, although the time complexity is still quadratic in the worst case scenario.