Explain the tournament sort algorithm.

The tournament sort algorithm is a comparison-based sorting algorithm that works by dividing the input array into smaller subarrays and repeatedly selecting the minimum element from each subarray to form a sorted sequence.

Here is a step-by-step explanation of the tournament sort algorithm:

1. Divide the input array into smaller subarrays: Start by dividing the input array into multiple subarrays, each containing a single element. If the input array has n elements, there will be n subarrays initially.

2. Construct a tournament tree: A tournament tree is a binary tree where each internal node represents a comparison between two elements, and each leaf node represents an element from the input array. The tree is constructed by comparing the elements in pairs and selecting the smaller element as the parent node. Repeat this process until there is only one element left, which becomes the root of the tree.

3. Find the minimum element: Traverse the tournament tree from the root to the leaves, always selecting the smaller child node at each internal node. This process will identify the minimum element in the input array.

4. Replace the minimum element: Once the minimum element is found, replace it with the next element from its corresponding subarray. If the subarray is exhausted, replace it with a sentinel value (e.g., infinity) to indicate that it has been fully processed.

5. Update the tournament tree: After replacing the minimum element, update the tournament tree by propagating the changes upwards. Compare the new element with its parent node and swap them if necessary to maintain the tournament property.

6. Repeat steps 3-5: Continue steps 3-5 until all subarrays are exhausted and the tournament tree is empty. At each iteration, the minimum element is selected and placed in the sorted sequence.

7. Merge the sorted sequences: Finally, merge the sorted sequences obtained from each subarray to obtain the fully sorted array.

The tournament sort algorithm has a time complexity of O(n log n), where n is the number of elements in the input array. This makes it an efficient sorting algorithm for large datasets. However, it requires additional space for constructing the tournament tree, resulting in a space complexity of O(n).