Overview
Shell Sort, named after its inventor Donald Shell, is an in-place comparison-based sorting algorithm. It generalizes insertion sort by allowing the exchange of items that are far apart. The main idea is to arrange the list of elements so that, starting anywhere, taking every h-th element (for some integer h) yields a sorted list. Such a list is said to be h-sorted. After the initial h-sorting pass, h is reduced, and the process is repeated until h equals 1.
Key Characteristics
- Time Complexity: Average case is O(n^(3/2)), worst-case can go up to O(n^2), and best-case is O(n log n).
- Space Complexity: O(1), as it is an in-place sorting algorithm.
- Stability: Not stable.
- Algorithm Type: Comparison-based sorting.
How Shell Sort Works
- Choose a Gap Sequence: Select an initial gap (e.g., hhh). A common sequence is to halve the gap until it becomes 1.
- h-sorting: Perform insertion sort for each gap, treating elements that are h positions apart as a sublist.
- Reduce the Gap: Reduce the gap and repeat the sorting process.
- Repeat: Continue until the gap is 1 and perform the final insertion sort.
Steps and Example
Consider the following unsorted array:
[12, 34, 54, 2, 3]Step 1: Choose a Gap Sequence
- Start with a gap of n/2n/2n/2 where nnn is the length of the array. For this example, the initial gap is 2.
Step 2: h-sorting
- Perform insertion sort for each sublist created by the gap.
Step 3: Reduce the Gap
- Reduce the gap by half and repeat the process.
Step 4: Final Insertion Sort with Gap 1
- Perform insertion sort with a gap of 1.
Visualization
Let’s generate a step-by-step visualization of Shell Sort using diagrams.
Initial Array
[12, 34, 54, 2, 3]Step 1: Initial Gap (h = 2)
Gap = 2
Sublists: [12, 54, 3] and [34, 2]Step 2: Perform Insertion Sort on Sublists
Sublists after sorting: [3, 12, 54] and [2, 34]
Merged array: [3, 2, 12, 34, 54]Step 3: Reduce the Gap (h = 1)
Gap = 1
Sublists: [3], [2], [12], [34], [54]Step 4: Perform Insertion Sort on Entire Array
Final sorted array: [2, 3, 12, 34, 54]