Shell Sort

• requires O(N^3/2) operations in the worst case, which means that it can be quite effectively used even for moderately large files (say N < 5000).
• Invented by Donald Shell in 1959, the shell sort is the most efficient of the O(n2) class of sorting algorithms.
• It is also the most complex of the O(n2) algorithms.
• The shell sort is a "diminishing increment sort", better known as a "comb sort" to the unwashed programming masses.

Pros: Efficient for medium-size lists.

Cons: Somewhat complex algorithm, not nearly as efficient as the merge, heap, and quick sorts.

The algorithm makes multiple passes through the list, and each time sorts a number of equally sized sets using the insertion sort. The size of the set to be sorted gets larger with each pass through the list, until the set consists of the entire list. (Note that as the size of the set increases, the number of sets to be sorted decreases.) This sets the insertion sort up for an almost-best case run each iteration with a complexity that approaches O(n).

Example and Steps in Shell Sort:

The idea is to divide the sequence into groups which induces great order into the initial random sequence relatively fast and then apply insertion sort to get final order.

To compare element which are relatively great distance from each other,

Steps:

1. half the sequence (4 elements/group), and compare corresponding elements (four groups, two elements/group).

Empirical Analysis

Shell Sort Efficiency

The shell sort is by far the fastest of the N2 class of sorting algorithms. It's more than 5 times faster than the bubble sort and a little over twice as fast as the insertion sort, its closest competitor.

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 is hyper-critical. It's also an excellent choice for repetitive sorting of smaller lists.

Source Code

Below is the basic shell sort algorithm.

void shellSort(int numbers[], int array_size)

{

int i, j, increment, temp;

increment = 3;

while (increment > 0)

{

for (i=0; i <>

{

j = i;

temp = numbers[i];

while ((j >= increment) && (numbers[j-increment] > temp))

{

numbers[j] = numbers[j - increment];

j = j - increment;

}

numbers[j] = temp;

}

if (increment/2 != 0)

increment = increment/2;

else if (increment == 1)

increment = 0;

else

increment = 1;

}

}

Selection Sort

• simple O(n²) sorting algorithm
• the method requires O(N2) comparisons and so it should only be used on small files.
• There is an important exception to this rule. When sorting files with large records and small keys, the cost of exchanging records controls the running time.
• In such cases, selection sort requires O(N) time since the number of exchanges is at most N.
• The selection sort works by selecting the smallest unsorted item remaining in the list, and then swapping it with the item in the next position to be filled. The selection sort has a complexity of O(n2).

Pros: Simple and easy to implement.

Cons: Inefficient for large lists, so similar to the more efficient insertion sort that the insertion sort should be used in its place.

Steps in Selection Sort:

1. Select the largest element of an array.

2. Swap the last element with the largest element.

3. Now do the same thing again, for an array that has one less element.

4. Keep doing this, until you are down to an array of 2 elements.

Example:

Repeatedly find the next largest (or smallest) element in the array and move it to its final position in the sorted array.

Empirical Analysis

Selection Sort Efficiency

The selection sort is the unwanted stepchild of the n2 sorts. It yields a 60% performance improvement over the bubble sort, but the insertion sort is over twice as fast as the bubble sort and is just as easy to implement as the selection sort. In short, there really isn't any reason to use the selection sort - use the insertion sort instead.

If you really want to use the selection sort for some reason, try to avoid sorting lists of more than a 1000 items with it or repetitively sorting lists of more than a couple hundred items.

Source Code

Below is the basic selection sort algorithm.

void selectionSort(int numbers[], int array_size)

{

int i, j;

int min, temp;

for (i = 0; i <>

{

min = i;

for (j = i+1; j <>

{

if (numbers[j] <>

min = j;

}

temp = numbers[i];

numbers[i] = numbers[min];

numbers[min] = temp;

}

}

Bubble Sort

· is the oldest and simplest sort in use.

· Unfortunately, it's also the slowest.

· The bubble sort is generally considered to be the most inefficient sorting algorithm in common usage. Under best-case conditions (the list is already sorted), the bubble sort can approach a constant O(n) level of complexity. General-case is an abysmal O(n2).

· While the insertion, selection, and shell sorts also have O(n2) complexities, they are significantly more efficient than the bubble sort.

Pros: Simplicity and ease of implementation.

Cons: Horribly inefficient.

Steps in Bubble Sort:

1. Comparing each item in the list with the item next to it

2. Swap them if required (if not in correct order).

3. The algorithm repeats this process until it makes a pass all the way through the list without swapping any items (in other words, all items are in the correct order).

4. This causes larger values to "bubble" to the end of the list while smaller values "sink" towards the beginning of the list.

In summary, each complete pass through the list has the effect of pushing each value slightly closer to it’s sorted position. After a finite number of passes (at most N-1) the list must eventually become completely sorted.

Example:

Consider this example list, below. Let’s use bubble sort to order it in ascending order.

4 8 1 9 3

First we would compare 4 and 8. They are in the correct order. So we move to the next pair, 8 and 1. They are not in the correct order, so they are swapped. The list now looks like:

4 1 8 9 3

Now we compare 8 and 9. They are in the correct order, so we move on to compare 9 and 3, which are not correctly ordered, and so are swapped. The list now looks like like:

4 1 8 3 9

That concludes one single pass. We now start again from the beginning, for the second pass. 4 and 1 are out of order, so they are swapped, and the list becomes:

1 4 8 3 9

4 and 8 are already correct, so they are left as is. 8 and 3 however need to be swapped, so the list becomes:

1 4 3 8 9

Note that we don’t bother comparing 8 and 9 on the second pass, because we know they are already in the right order. Thus we now move straight to the next pass. By comparing 1 and 4, we find that they are correctly ordered.

But 4 and 3, the next pair, are not and so are swapped. This results in the list:

1 3 4 8 9

It can be seen quite obviously that this small example is now sorted. However, the bubble sort algorithm presented would still perform the next pass, obviously to no effect.

Empirical Analysis

Bubble Sort Efficiency

The graph clearly shows the n² nature of the bubble sort.

A fair number of algorithm purists (which means they've probably never written software for a living) claim that the bubble sort should never be used for any reason. Realistically, there isn't a noticeable performance difference between the various sorts for 100 items or less, and the simplicity of the bubble sort makes it attractive. The bubble sort shouldn't be used for repetitive sorts or sorts of more than a couple hundred items.

Source Code

Below is the basic bubble sort algorithm.

void bubbleSort(int numbers[], int array_size)

{

int i, j, temp;

for (i = (array_size - 1); i >= 0; i--)

{

for (j = 1; j <= i; j++)

{

if (numbers[j-1] > numbers[j])

{

temp = numbers[j-1];

numbers[j-1] =numbers[j];

numbers[j] = temp;

}
}
}
}

Insertion Sort

• is a most natural and general one
• Insertion Sort orders a list in the same way we would order a hand of playing cards.
• its worst case and average behavior analysis are easy.
• Using Insertion Sort, the number of comparisons on a list of size n varies, because, if the numbers are already in order, further comparisons are avoided. So we will find the average number of comparisons.
• Using limits and probability, we find that the average number of comparisons is (n-1)(n/4) + k, where k increases for larger values of n.

Pros: Relatively simple and easy to implement.
Cons: Inefficient for large lists.

The insertion sort works just like its name suggests - it inserts each item into its proper place in the final list. The simplest implementation of this requires two list structures - the source list and the list into which sorted items are inserted. To save memory, most implementations use an in-place sort that works by moving the current item past the already sorted items and repeatedly swapping it with the preceding item until it is in place.

Like the bubble sort, the insertion sort has a complexity of O(n²). Although it has the same complexity, the insertion sort is a little over twice as efficient as the bubble sort.

Steps in Insertion Sort:
1. Compare the first two numbers, placing the smallest one in the first position.
2. Compare the third number to the second number. If the third number is larger, then the first three numbers are in order. If not, then swap them.
3. Now compare the numbers in positions one and two and swap them if necessary.
4. Proceed in this manner until reaching the end of the list.

Example:

Empirical Analysis

Insertion Sort Efficiency

The graph demonstrates the n² complexity of the insertion sort.

The insertion sort is a good middle-of-the-road choice for sorting lists of a few thousand items or less. The algorithm is significantly 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 and almost 40% faster than the selection 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.

Source Code

Below is the basic insertion sort algorithm.

void insertionSort(int numbers[], int array_size)

{

int i, j, index;

for (i=1; i < style="">

{

index = numbers[i];

j = i;

while ((j > 0) && (numbers[j-1] > index))

{

numbers[j] = numbers[j-1];

j = j - 1;

}

numbers[j] = index;

}

}