The library sort algorithm is an efficient sorting technique that merges the advantages of insertion sort and binary search.

Library sort, also referred to as gapped insertion sort, was developed by Michael A. Bender and Martin Farach-Colton in 2004. This algorithm operates by maintaining a dynamic array that contains gaps, or free spaces, between elements. These gaps facilitate the efficient insertion of new elements during the sorting process.

The algorithm begins by inserting the first element of the input list into the dynamic array. For each subsequent element, it employs binary search to determine the appropriate position within the sorted section of the array. If the identified position is already occupied, the algorithm shifts the existing elements to the right until a free space is located, and the new element is inserted there.

A crucial aspect of the library sort’s efficiency is its management of gaps. The algorithm ensures that there are always enough gaps evenly distributed throughout the array. When the array becomes too full, meaning there are insufficient gaps, the algorithm increases the array’s size and redistributes the elements to restore the necessary gaps.

In terms of performance, the average-case time complexity of library sort is $O(n \log n)$, which is comparable to that of quicksort, heapsort, and mergesort. However, it is important to note that library sort has a higher space complexity due to the requirement for additional gaps, making it less suitable for memory-constrained environments.

In summary, library sort is an innovative algorithm that effectively combines the strengths of insertion sort and binary search. By maintaining a dynamic array with gaps, it allows for efficient insertions while using binary search to quickly locate the correct position for each element. This results in a fast and efficient sorting mechanism, albeit with increased memory usage.