Sorting Techniques

Insertion Sort &
Shellsort
By: Andy Le
CS146 – Dr. Sin Min Lee
Spring 2004

Outline
 Importance of Sorting
 Insertion Sort
 Explanation
 Runtime
 Advantage and Disadvantage
 Walk through example
 Shell Sort
 History
 Explanation
 Runtime
 Advantage and Disadvantage
 Walk through example

Why we do sorting?
 Commonly encountered programming task in
computing.
 Examples of sorting:
 List containing exam scores sorted from Lowest to
Highest or from Highest to Lowest
 List containing words that were misspelled and be
listed in alphabetical order.
 List of student records and sorted by student
number or alphabetically by first or last name.

Why we do sorting?

 Searching for an element in an array will
be more efficient. (example: looking up
for information like phone number).
 It’s always nice to see data in a sorted
display. (example: spreadsheet or
database application).
 Computers sort things much faster.

History of Sorting

 Sorting is one of the most important
operations performed by computers. In
the days of magnetic tape storage before
modern databases, database updating
was done by sorting transactions and
merging them with a master file.

History of Sorting

 It's still important for presentation of data
extracted from databases: most people
prefer to get reports sorted into some
relevant order before flipping through
pages of data!

Insertion Sort
 Insertion sort keeps making the left side of the
array sorted until the whole array is sorted. It
sorts the values seen far away and repeatedly
inserts unseen values in the array into the left
sorted array.
 It is the simplest of all sorting algorithms.
 Although it has the same complexity as Bubble
Sort, the insertion sort is a little over twice as
efficient as the bubble sort.

Insertion Sort

 Real life example:
 An example of an insertion sort occurs in
everyday life while playing cards. To sort the
cards in your hand you extract a card, shift
the remaining cards, and then insert the
extracted card in the correct place. This
process is repeated until all the cards are in
the correct sequence.

Insertion Sort runtimes

 Best case: O(n). It occurs when the data
is in sorted order. After making one pass
through the data and making no
insertions, insertion sort exits.
 Average case: θ(n^2) since there is a
wide variation with the running time.
 Worst case: O(n^2) if the numbers were
sorted in reverse order.

Empirical Analysis of Insertion Sort

The graph demonstrates the n^2 complexity of the insertion sort.

Source: http://linux.wku.edu/~lamonml/algor/sort/insertion.html

Insertion Sort

 The insertion sort is a good choice for
sorting lists of a few thousand items or
less.

Insertion Sort

 The insertion sort shouldn't be used for
sorting lists larger than a couple
thousand items or repetitive sorting of
lists larger than a couple hundred items.

Insertion Sort

 This algorithm is much simpler than the
shell sort, with only a small trade-off in
efficiency. At the same time, the insertion
sort is over twice as fast as the bubble
sort.

Advantage of Insertion Sort

 The advantage of Insertion Sort is that it
is relatively simple and easy to
implement.

Disadvantage of Insertion Sort

 The disadvantage of Insertion Sort is that
it is not efficient to operate with a large
list or input size.

Insertion Sort Example

 Sort: 34 8 64 51 32 21
 34 8 64 51 32 21
 The algorithm sees that 8 is smaller than 34
so it swaps.
 8 34 64 51 32 21
 51 is smaller than 64, so they swap.
 8 34 51 64 32 21

Insertion Sort Example
 Sort: 34 8 64 51 32 21
 8 34 51 64 32 21 (from previous slide)
 The algorithm sees 32 as another smaller
number and moves it to its appropriate
location between 8 and 34.
 8 32 34 51 64 21
 The algorithm sees 21 as another smaller
number and moves into between 8 and 32.
 Final sorted numbers:
 8 21 32 34 51 64

Shellsort
 Founded by Donald Shell and named the
sorting algorithm after himself in 1959.
 1st algorithm to break the quadratic time barrier
but few years later, a sub quadratic time bound
was proven
 Shellsort works by comparing elements that
are distant rather than adjacent elements in an
array or list where adjacent elements are
compared.

Shellsort

 Shellsort uses a sequence h1, h2, …, ht
called the increment sequence. Any
increment sequence is fine as long as h 1
= 1 and some other choices are better
than others.

Shellsort

 Shellsort makes multiple passes through
a list and sorts a number of equally sized
sets using the insertion sort.

Shellsort

 Shellsort improves on the efficiency of
insertion sort by quickly shifting values to
their destination.

Shellsort

 Shellsort is also known as diminishing
increment sort.
 The distance between comparisons
decreases as the sorting algorithm runs
until the last phase in which adjacent
elements are compared

Shellsort

 After each phase and some increment
hk, for every i, we have a[ i ] ≤ a [ i + hk ]
all elements spaced hk apart are sorted.
 The file is said to be hk – sorted.

Empirical Analysis of Shellsort

Source: http://linux.wku.edu/~lamonml/algor/sort/shell.html

(Advantage)
 Advantage of Shellsort is that its only
efficient for medium size lists. For bigger
lists, the algorithm is not the best choice.
Fastest of all O(N^2) sorting algorithms.
 5 times faster than the bubble sort and a
little over twice as fast as the insertion
sort, its closest competitor.

(Disadvantage)
 Disadvantage of Shellsort is that it is a complex
algorithm and its not nearly as efficient as the
merge, heap, and quick sorts.
 The shell sort is still significantly slower than
the merge, heap, and quick sorts, but its
relatively simple algorithm makes it a good
choice for sorting lists of less than 5000 items
unless speed important. It's also an excellent
choice for repetitive sorting of smaller lists.

Shellsort Best Case

 Best Case: The best case in the shell
sort is when the array is already sorted in
the right order. The number of
comparisons is less.

Shellsort Worst Case

 The running time of Shellsort depends on
the choice of increment sequence.
 The problem with Shell’s increments is
that pairs of increments are not
necessarily relatively prime and smaller
increments can have little effect.

Shellsort Examples
 Sort: 18 32 12 5 38 33 16 2
8 Numbers to be sorted, Shell’s increment will be floor(n/2)
* floor(8/2)  floor(4) = 4

increment 4: 1 2 3 4 (visualize underlining)

18 32 12 5 38 33 16 2

Step 1) Only look at 18 and 38 and sort in order ;
18 and 38 stays at its current position because they are in order.
2 and 5 need to be switched to be in order.

Shellsort Examples (con’t)
 Sort: 18 32 12 5 38 33 16 2
Resulting numbers after increment 4 pass:
18 32 12 2 38 33 16 5
* floor(4/2)  floor(2) = 2
increment 2: 1 2

18 32 12 2 38 33 16 5
Step 1) Look at 18, 12, 38, 16 and sort them in their appropriate location:

12 38 16 2 18 33 38 5
Step 2) Look at 32, 2, 33, 5 and sort them in their appropriate location:
12 2 16 5 18 32 38 33

Shellsort Examples (con’t)
 Sort: 18 32 12 5 38 33 16 2
* floor(2/2)  floor(1) = 1
increment 1: 1
12 2 16 5 18 32 38 33

2 5 12 16 18 32 33 38

The last increment or phase of Shellsort is basically an Insertion
Sort algorithm.

Additional Online References

 Spark Notes (From Barnes & Noble):
 http://www.sparknotes.com/cs/

Sorting Techniques

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Sorting Techniques