SHELL SORT

Shell Sort

Chapter-4

Bucket sort

Bucket sort example

Bucket sort Example 2

Shell Sort

• Shell sort works by comparing elements that are distant rather than adjacent elements in an array.

• Shell sort improves on the efficiency of insertion sort by quickly shifting values to their destination.

• Shell sort 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

Shell Sort

• 18 32 12 5 38 33 16 2floor(8/2) floor(4) = 418 32 12 5 38 33 16 2

Resulting numbers after increment 4 pass:

18 32 12 2 38 33 16 5

Shell Sort

floor(4/2) floor(2) = 2

increment 2:

• 18 32

• 12 2

• 38 33

• 16 5

• Again sort the column….

Shell Sort

• The result is,

• 12 2

• 16 5

• 18 32

• 38 33

Shell Sort

floor(2/2) floor(1) = 1

• 12 2 16 5 18 32 38 33

• The last increment or phase of Shell sort is basically an Insertion Sort algorithm.

2 5 12 16 18 32 33 38

Shell Sort

• Arrange the list into a table and sort the columns (using an insertion sort).

• Repeat this process, each time with smaller number of longer columns.

• At the end, the table has only one column.

• For example, consider a list of numbers like • [ 13 14 94 33 82 25 59 94 65 23 45 27 73 25 39 10 ]. • If we started with a step-size of 5, we could visualize

this as breaking the list of numbers into a table with 5 columns.

Example

• This would look like this:

• 13 14 94 33 82

• 25 59 94 65 23

• 45 27 73 25 39

• 10

Example Contd…

• We then sort each column, which gives us

• 10 14 73 25 23

• 13 27 94 33 39

• 25 59 94 65 82

• 45

Example Contd…

• When read back as a single list of numbers, we get [ 10 14 73 25 23 13 27 94 33 39 25 59 94 65 82 45 ].

• Here, the 10 which was all the way at the end, has moved all the way to the beginning.

• This list is then again sorted using a 3-gap sort as shown below.

Example Contd…

• 10 14 73

• 25 23 13

• 27 94 33

• 39 25 59

• 94 65 82

• 45

• Again sort each column

Example Contd…

• Which gives us • 10 14 13 • 25 23 33 • 27 25 59 • 39 65 73• 45 94 82• 94 • Now all that remains to be done is a 1-gap sort

(simple insertion sort).

INTERPOLATION SEARCH:

• It is a modified algorithm of Binary search reducing the complexity.

• Best/Average case complexity is O(log log n) ) which will occur when elements of the array are fairly uniformly distributed over their interval eg. . [3,5,8,10,13,15]

• Worst case complexity is O(n) which will occur when the elements are not fairly uniformly distributed over their interval eg. [3,5,8,10,13,100]

Example:

• Search for the key 20 in the following array using Interpolation Search

• int[] a = {0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30}

Interpolation Search:

• middle = low + ( high - low )*( (key – a[low]) / ( a[high] – a[low] ))

• middle = 0 + ceil(15 – 0)* ( (20 – 0) / (30 – 0) ) = 0 +10 =10

• a[middle] = 20 which is equal to key.

• So key found at index 10.