Compare the time complexities of shell sort, comb sort, and cycle sort.

Shell sort, comb sort, and cycle sort are all comparison-based sorting algorithms that have different time complexities.

1. Shell Sort:
Shell sort is an in-place comparison sort algorithm that improves upon the insertion sort algorithm. It works by sorting elements that are far apart and progressively reducing the gap between elements to be compared. The time complexity of Shell sort depends on the chosen gap sequence. The worst-case time complexity of Shell sort is O(n^2), but it can be improved to O(n log n) by using certain gap sequences like the Pratt sequence or the Sedgewick sequence.

2. Comb Sort:
Comb sort is also an in-place comparison sort algorithm that improves upon the bubble sort algorithm. It works by comparing elements that are far apart and gradually reducing the gap between elements. The time complexity of Comb sort is O(n^2) in the worst-case scenario. However, by using a shrink factor of 1.3, the average-case time complexity can be improved to O(n log n).

3. Cycle Sort:
Cycle sort is an in-place comparison sort algorithm that minimizes the number of writes to memory. It works by dividing the input into cycles and rotating the elements within each cycle until the correct position is reached. The time complexity of Cycle sort is O(n^2) in the worst-case scenario. However, it is considered to be efficient for small arrays or when the number of memory writes is a concern.

In summary, the time complexities of these sorting algorithms are as follows:
- Shell sort: O(n^2) in the worst-case, but can be improved to O(n log n) with certain gap sequences.
- Comb sort: O(n^2) in the worst-case, but O(n log n) on average with a shrink factor of 1.3.
- Cycle sort: O(n^2) in the worst-case, but efficient for small arrays or when minimizing memory writes is important.