How does the library sort algorithm work?

The library sort algorithm works by combining insertion sort and binary search to efficiently sort a list.

Library sort, also known as gapped insertion sort, is a sorting algorithm that combines the efficiency of insertion sort and binary search. It was invented by Michael A. Bender and Martin Farach-Colton in 2004. The algorithm works by maintaining a dynamic array with gaps, or free spaces, between elements. This allows for efficient insertion of new elements during the sorting process.

The algorithm begins by inserting the first element of the input into the dynamic array. Then, for each subsequent element, the algorithm uses binary search to find the correct position in the sorted part of the array, and inserts the element there. If the position is already occupied, the algorithm moves elements to the right until a free space is found. This is where the element is inserted.

The key to the efficiency of library sort is the maintenance of these gaps. The algorithm ensures that there is always a sufficient number of gaps distributed evenly throughout the array. If the array becomes too full (i.e., there are not enough gaps), the algorithm increases the size of the array and redistributes the elements to restore the gaps.

The time complexity of library sort is O(n log n) in the average case, which makes it as fast as quicksort, heapsort, and mergesort. However, library sort has a higher space complexity due to the need for extra gaps. This makes it less suitable for situations where memory is limited.

In summary, library sort is a clever algorithm that combines the best aspects of insertion sort and binary search. It maintains a dynamic array with gaps to allow for efficient insertions, and uses binary search to quickly find the correct position for each element. This results in a fast and efficient sorting algorithm, albeit at the cost of increased memory usage.