How does the cycle sort algorithm function?

Cycle sort is an in-place, unstable sorting algorithm that operates by repeatedly swapping elements until they are in their correct positions.

The cycle sort algorithm works by visualising the elements of the array as a cyclic graph, where each node represents an element and the edges represent the desired positions of the elements. The algorithm begins by identifying cycles in the graph and moving elements along these cycles until each element is in its correct position.

To identify a cycle, the algorithm starts at the first element in the array and calculates its correct position in the sorted array. If the element is already in the correct position, the algorithm moves on to the next element. If the element is not in the correct position, the algorithm swaps it with the element currently occupying its correct position. This process is repeated until the first element is back in its correct position, at which point one cycle is complete.

The algorithm then moves on to the next element in the array that has not yet been processed and identifies the next cycle. This process is repeated until all elements have been processed and are in their correct positions.

One of the key features of cycle sort is that it minimises the number of memory writes, which makes it useful in situations where write operations are costly. However, it is not a stable sort, which means that equal elements may not retain their relative order in the sorted array.

In terms of time complexity, cycle sort performs poorly compared to other sorting algorithms. It has a worst-case and average time complexity of O(n^2), which makes it inefficient for large data sets. However, its space complexity is O(1), which means it uses a constant amount of extra space regardless of the size of the input array.

In summary, cycle sort is a unique sorting algorithm that operates by identifying cycles in the array and moving elements along these cycles. While it is not the most efficient sorting algorithm in terms of time complexity, it has the advantage of minimising memory writes and using a constant amount of extra space.