Sorts

Tonga Institute of Higher Education

Introduction - 1

Sorting – The act of ordering data Often, we need to order data.

Example: Order a list of names

Different sorting techniques work best in different situations

Sorting is very important to making efficient programs so a lot of research is done in this area.

Introduction - 2

Sorting Algorithms Bubble Sort Selection Sort Insertion Sort

All these sorting routines do 2 steps repeatedly

1. Compare 2 items2. Swap 2 items or copy 1 item

Bubble Sort

Very Slow Simplest way to sort Rules

1. Start from left side2. Compare 2 items3. If the item on the left is bigger, switch them4. Move cursor one position right5. Repeat from step 2 until cursor is at the end of the

list. At this point, you have put the largest item at the end of the list. This item will no longer be included in this algorithm.

6. Repeat from step 1 until items are sorted

Demonstration

Bubble Sort Applet

Demonstration

Bubble Sort Code

Project: BubbleSort

Bubble Sort Efficiency Number of Comparisons

First Pass Number of items - 1 = N - 1

Second Pass Number of items - 2 = N – 2

So total is: (N -1) + (N – 2) + … + 1 = N*(N-1) / 2 = (N2 – N) / 2

Number of Swaps If the data is random,

swaps occur ½ the time when a comparison is made.

((N2 – N) / 2) / 2 = (N2 – N) / 4

Both comparisons and swaps have an N2 We ignore the constant numbers Therefore, the bubble sort compares and swaps in O(N2) time.

Selection Sort

Has less number of swaps than bubble sorts

Rules1. Check every item to find smallest value2. Swap the item with the smallest value with

the leftmost item. The leftmost item will no longer be included in this algorithm.

3. Repeat from step 1 until items are sorted

Demonstration

Selection Sort Applet

Demonstration

Selection Sort Code

Project: SelectionSort

Selection Sort Efficiency - 1 Number of Comparisons

First Pass Number of items - 1 = N - 1

Second Pass Number of items - 2 = N – 2

So total is: (N -1) + (N – 2) + (N – 3) + … + 1 = N*(N-1)/2

Number of Swaps Less than N swaps

Selection Sort Efficiency - 2 Number of Comparisons

Total is: = N*(N-1)/2

Number of Swaps Total is:

Less than N

Comparisons have an N2 We ignore the constant numbers Therefore, the selection sort compares in O(N2) time

Swaps have an N Therefore, the selection sort swaps in O(N) time

For small N values: Selection sorts run faster than bubble sorts because there are less swaps.

For large N values: Selection sorts run faster than bubble sorts because there are less swaps. However, the performance will be close to O(N2) than O(N) because the comparison times will dominate the swap times.

Insertion Sort - 1

Partial Sorting – Keeping a sorted list of items in a unsorted list

Marked Item – The leftmost item that has not been sorted

Insertion Sort - 2 Rules

1. Take out marked item

2. Compare marked item with item on left

3. If the item on left is bigger, then move bigger item 1 place to the right

4. Repeat from step 2 until the item on the left is smaller. Then, put the marked item in the blank space

5. Set marked item 1 place to the right. Repeat from step 1

Demonstration

Insertion Sort Applet

Demonstration

Insertion Sort Code

Project: InsertionSort

Insertion Sort Efficiency - 1 Number of Comparisons

First Pass 1

Second Pass 2

Last Pass N - 1

So total is: 1 + 2 + … + N - 1 = N*(N-1)/2

However, on average, only half of the items are compared before the insertion point is found.

So the total: = N*(N-1)/4

Number of Copies Almost the same as the number of comparisons

Insertion Sort Efficiency - 2 Number of Comparisons

Total: = N*(N-1)/4

Number of Copies Total:

= N*(N-1)/4

Comparisons have an N2 Therefore, the insertion sort compares in O(N2) time However, does half as many comparisons as bubble sorts and selection sorts

Copies have an N2

Therefore, the insertion sort copies in O(N2) time A copy is much faster than a swap This is twice as fast as a bubble sort for random data This is slightly faster than a selection sort for random data This is much faster when data is almost sorted This is the same as a bubble sort for inverse sorted order because every

possible comparison must be carried out

Comparing the 3 Sorts

Bubble Sorts Easiest Slowest

Selection Sorts Same number of comparisons as bubble sorts Much less number of swaps than bubble sorts Works well when data is small and swapping takes much longer

than comparisons Insertion Sorts

Best sort we’ve learned about ½ the number of comparisons as bubble sorts and selection

sorts Copies instead of swaps, which is much faster Works well when data is small and data is almost sorted