Lecture 23: Sorting 3 – Shell Sort

Objectives of this lecture

q Lean the weakness of Insertion Sorting Method

q Lear an improvement of insertion sorting method – shell sort.

Problem of Insertion Sort:

q There are actually two things that are considered when evaluating the efficiency of a sorting algorithm: number of comparisons and the number of moves (swapping of elements).

q As explained earlier, the number of comparisons required in both selection sort and insertion sort is about n*n except for an ordered list in which the insertion sort requires about n comparisons.

q However, if we consider the number of moves involved - which can be very tasking especially when the data items are records, insertion sort turn out to be worst than even selection sort.

q In the case of selection sort, we only need to make one swap per iteration (swap the minimum element with the one at the beginning).

q In the case of insertion sort, each time we pick an element from the source sub-list, we have to move all the elements in the target sub-list that are bigger than the element to be inserted, to the next cell.

q For example, the following table shows the number of moves required for a list of 13 elements:

Original:	28	81	36	47	17	13	55	65	23	18	67	38	3	Moves
step1:	28	81	36	47	17	13	55	65	23	18	67	38	3	0
step2:	28	36	81	47	17	13	55	65	23	18	67	38	3	1
step3:	28	36	47	81	17	13	55	65	23	18	67	38	3	1
step4:	17	28	36	47	81	13	55	65	23	18	67	38	3	4
step5:	13	17	28	36	47	81	55	65	23	18	67	38	3	5
step6:	13	17	28	36	47	55	81	65	23	18	67	38	3	1
step7:	13	17	28	36	47	55	65	81	23	18	67	38	3	1
step8:	13	17	23	28	36	47	55	65	81	18	67	38	3	6
step9:	13	17	18	23	28	36	47	55	65	81	67	38	3	7
step10:	13	17	18	23	28	36	47	55	65	67	81	38	3	1
step11:	13	17	18	23	28	36	38	47	55	65	67	81	3	5
step12:	3	13	17	18	23	28	36	38	47	55	65	67	81	12

Total number of moves = 44

Shell Sort

q This is a simple variation of insertion sort that can dramatically improve the processing time

q The aim of the shell sort, (named after its inventor, Donald Shell, 1959) is to reduce the number of moves required in insertion sort.

q When looking for a place to insert an item, shell sort first look at an element that is some distance d from the right hand side of the target sub-list. d for example can be initially ½ or 1/3 of the list size.

q On succeeding passes, d is reduced by some factor and the sorting is re-applied.

q By the time d reduces to one (equivalent do the normal insert sort) the items gets nearly sorted and the number of moves is drastically reduced.

q The implementation is as follows:

void shell_sort(int list[])

{ int d=MAX_SIZE;

do

{ d = d/3+1;

ins_sort2(list,d);

} while (d>1);

}

void ins_sort2(int list[], int d)

{ int i, j, temp;

for (i=d; i<MAX_SIZE; i++)

{ temp=list[i];

j=i;

while ((j-d>=0) && list[j-d]>temp)

{ list[j]=list[j-d];

j=j-d;

}

list[j]=temp;

}

}

q The following table shows the trace of the shell sort function:

Original:	28	81	36	47	17	13	55	65	23	18	67	38	3	Moves
For d=5
step1:	13	81	36	47	17	28	55	65	23	18	67	38	3	1
step2:	13	55	36	47	17	28	81	65	23	18	67	38	3	1
step3:	13	55	36	47	17	28	81	65	23	18	67	38	3	0
step4:	13	55	36	23	17	28	81	65	47	18	67	38	3	1
step5:	13	55	36	23	17	28	81	65	47	18	67	38	3	0
step6:	13	55	36	23	17	28	81	65	47	18	67	38	3	0
step7:	13	38	36	23	17	28	55	65	47	18	67	81	3	2
step8:	13	38	3	23	17	28	55	36	47	18	67	81	65	2
For d=2
step1:	3	38	13	23	17	28	55	36	47	18	67	81	65	1
step2:	3	23	13	38	17	28	55	36	47	18	67	81	65	1
step3:	3	23	13	38	17	28	55	36	47	18	67	81	65	0
step4:	3	23	13	28	17	38	55	36	47	18	67	81	65	1
step5:	3	23	13	28	17	38	55	36	47	18	67	81	65	0
step6:	3	23	13	28	17	36	55	38	47	18	67	81	65	1
step7:	3	23	13	28	17	36	47	38	55	18	67	81	65	1
step8:	3	18	13	23	17	28	47	36	55	38	67	81	65	4
step9:	3	18	13	23	17	28	47	36	55	38	67	81	65	0
step10:	3	18	13	23	17	28	47	36	55	38	67	81	65	0
step11:	3	18	13	23	17	28	47	36	55	38	65	81	67	1
For d= 1
step1:	3	18	13	23	17	28	47	36	55	38	65	81	67	0
step2:	3	13	18	23	17	28	47	36	55	38	65	81	67	1
step3:	3	13	18	23	17	28	47	36	55	38	65	81	67	0
step4:	3	13	17	18	23	28	47	36	55	38	65	81	67	2
step5:	3	13	17	18	23	28	47	36	55	38	65	81	67	0
step6:	3	13	17	18	23	28	47	36	55	38	65	81	67	0
step7:	3	13	17	18	23	28	36	47	55	38	65	81	67	1
step8:	3	13	17	18	23	28	36	47	55	38	65	81	67	0
step9:	3	13	17	18	23	28	36	38	47	55	65	81	67	2
step10:	3	13	17	18	23	28	36	38	47	55	65	81	67	0
step11:	3	13	17	18	23	28	36	38	47	55	65	81	67	0
step12:	3	13	17	18	23	28	36	38	47	55	65	67	81	1

Total number of moves = 24