Shell sort is a generalization of insertion sort that improves efficiency by allowing exchanges of elements that are farther apart. It works by making multiple passes through the list with decreasing increments to improve on insertion sort. Radix sort is a non-comparative sorting algorithm that sorts integer keys by grouping them based on the place value of their digits. It has linear time complexity, making it very fast for sorting integers compared to other algorithms. While fast for integers, radix sort requires additional space and only works for integer keys.

1. SHELL SORT ALGORITHM

2. Shell Sort
▶ Founded by Donald Shell in 1959
▶ 1st algorithm to break the quadratic time barrier
▶ a highly efficient sorting algorithm
▶ known as diminishing increment sort
▶ Generalization of insertion short

3. How does Shell Sort Works?
▶ works by comparing elements that are distant rather than
adjacent elements in an array or list where adjacent elements are
compared
▶ uses increment sequence
▶ makes multiple passes through a list
▶ improves on the efficiency of insertion sort
▶ decreases distance between comparisons

4. Example
Here these values are {35, 14}, {33, 19}, {42, 27} and {10, 44}

5. Example
Compare values in each sub-list and swap them (if necessary) in the
original array. After this step, the new array should look like this −

6. Example
Then, take interval of 2 and this gap generates two sub-lists - {14, 27,
35, 42}, {19, 10, 33, 44}

7. Example
Again, compare and swap the values, if required, in the original array.
After this step, the array should look like this −
Finally, the rest of the array sorted using interval
of value 1. Shell sort uses insertion sort to sort the
array.

8. Example
We see that it
required only four
swaps to sort the rest
of the array.

9. Empirical Analysis of Shell Sort

10. Empirical Analysis of Shell Sort
(Advantage)
▶ Fastest of all O (N^2) sorting algorithms
▶ it’s only efficient for medium size lists
▶ Five (5) times faster than the bubble sort
▶ a little over twice as fast as the insertion sort

11. Empirical Analysis of Shell Sort
(Dis-advantage)
▶ it is a complex algorithm
▶ it’s not nearly as efficient as the merge, heap, and quick sorts
▶ still significantly slower than the merge, heap, and quick sorts
▶ also an excellent choice for repetitive sorting of smaller lists.

12. Shell Sort Best Case
▶ when the array is already sorted in the right order
▶ the number of comparisons is less.

13. Shell Sort Worst Case
▶ running time of Shell sort depends on the choice of increment
sequence.
▶ pairs of increments are not necessarily relatively prime and smaller
increments can have little effect.

14. RADIX SORT ALGORITHM

15. Radix Sort
▶ Founded by Harold H. Seward in 1954.
▶ Radix sort is a non-comparative integer sorting algorithm that sorts
data with integer keys by grouping keys by the individual digits which
share the same significant position and value.
▶ Unlike other sorting methods, radix sort considers the structure of the
keys
▶ Sorting is done by comparing bits in the same position
▶ Extension to keys that are alphanumeric strings

16. How does Radix Sort Works?
▶ sorts by grouping numbers by their individual digits (or by
their radix).
▶ It uses each radix digit as a key, and implements counting sort or
bucket sort under the hood in order to do the work of sorting.
▶ Take the least significant digit of each key.
▶ Group the keys based on that digit, but otherwise keep the original
order of keys.
▶ Repeat the grouping process with each more significant digit.

17. Example
Original, unsorted list:
170, 45, 75, 90, 802, 24, 2, 66

20. Complexity
▶ The complexity of radix sort is linear, which in terms of omega means
O(n). That is a great benefit in performance compared to O(n.log(n))
or even worse with O(n2) as we can see on the following chart.

21. Why using radix sort?
1. It’s fast
Radix sort is very fast compared to other sorting algorithms as we saw on
the diagram above. This algorithm is very useful in practice because in
practice we often sort sets of integers.
2. It’s easy to understand and implement
Even a beginner can understand and implement radix sort, which is
great. You need no more than few loops to implement it.

22. Why NOT using radix sort?
1. Works only with integers
If you’re not sure about the input better do not use radix sort. We may
think that our input consists only of integers and we can go for radix
sort, but what if in the future someone passes floats or strings to our
routine.
2. Requires additional space
Radix sort needs additional space – at least as much as the input.

23. Empirical Analysis of Radix Sort

24. Final Words
▶ Radix sort is restricted by the input’s domain, but I must say that in
practice there are tons of cases where only integers are sorted. This is
when we get some data from the based on primary keys – typically
primary in database tables are integers as well. So practically there
are lots of cases of sorting integers, so radix sort may be one very,
very useful algorithm and it is so cool that it is also easy to
implement.