How does tim sort algorithm sort elements?

Tim sort algorithm sorts elements by dividing the input into chunks, sorting them individually, and then merging them in a specific order.

Tim sort is a hybrid sorting algorithm, derived from merge sort and insertion sort, designed to perform well on many kinds of real-world data. It was implemented by Tim Peters in 2002 for use in the Python programming language. The algorithm finds subsequences of the data that are already ordered, and uses the patterns to sort the remainder more efficiently. This is done by dividing the input into chunks, known as 'runs', which are then sorted individually using insertion sort.

The runs are chosen in a specific way. If the run is in ascending order, it is extended as far as possible. If it is in descending order, it is reversed. If the run is shorter than a certain length, it is extended by binary insertion sort. This process of creating runs is what makes Tim sort adaptive, as it takes advantage of any existing order in the data.

Once the runs are sorted, they are merged together. This is done in a specific order, controlled by a stack of pending runs. The merging process is similar to that of merge sort, but with a few modifications to improve performance. For example, the algorithm keeps track of the sizes of the runs on the stack, and ensures that certain invariants are maintained. These invariants are designed to ensure that the total time spent merging is linear, regardless of the distribution of run lengths.

In summary, Tim sort is a sophisticated algorithm that combines the best features of insertion sort and merge sort. It is adaptive, stable, and has a worst-case and average time complexity of O(n log n). It is particularly effective on data that is already partially ordered, which is common in many real-world scenarios.