h = 1 while h < n, h = 3*h + 1 while h > 0, h = h / 3 for k = 1:h, insertion sort a[k:h:n]→ invariant: each h-sub-array is sortedend

- Not stable
- O(1) extra space
- O(n
^{3/2}) time as shown (see below) - Adaptive: O(n·lg(n)) time when nearly sorted

The worse-case time complexity of shell sort depends on the
increment sequence. For the increments *1 4 13 40 121...*,
which is what is used here, the time complexity
is O(n^{3/2}). For other increments, time
complexity is known to be O(n^{4/3}) and even
O(n·lg^{2}(n)). Neither tight upper bounds
on time complexity nor the best increment sequence are known.

Because shell sort is based on insertion sort, shell sort
inherits insertion sort's adaptive properties. The
adapation is not as dramatic because shell sort requires one
pass through the data for each increment, but it is
significant. For the increment sequence shown above, there
are log_{3}(n) increments, so the time complexity
for nearly sorted data is O(n·log_{3}(n)).

Because of its low overhead, relatively simple implementation, adaptive properties, and sub-quadratic time complexity, shell sort may be a viable alternative to the O(n·lg(n)) sorting algorithms for some applications when the data to be sorted is not very large.

- Black values are sorted.
- Gray values are unsorted.
- Dark gray values show the current sub-array that is being sorted using insertion sort.
- A red triangle marks the algorithm position.