Analysis of Algorithm (Quicksort & Bubblesort)

1. Quick Sort
Jimenez, Jake
Rojas, Davis Jacob
Miguel, Flynce

2. Quick Sort
 Quicksort uses divide-and-conquer, and so it's a recursive algorithm. The
way that quicksort uses divide-and-conquer is a little different from how
merge sort does. In merge sort, the divide step does hardly anything, and
all the real work happens in the combine step.
 Quicksort is the opposite: all the real work happens in the divide step. In
fact, the combine step in quicksort does absolutely nothing.

3. Quick Sort
 Here is how quicksort uses divide-and-conquer: think of sorting a sub
array[p…r], Where initially the sub array is array[0…n-1].
We take this set of array as an example:
40 20 10 80 60 50 7 30 100
0 1 2 3 4 5 6 7 8

4. Quick Sort
 Divide
1. Choose any element in the sub array [p…r] call this element Pivot.
We now have our Pivot which is 40 at array 0
40 20 10 80 60 50 7 30 100
0 1 2 3 4 5 6 7 8

5. Quick Sort
 Given a pivot, re-arrange the elements of the array such that the resulting array
consists of:
 One sub-array that contains elements >= pivot
 Another sub-array that contains elements < pivot
We call this procedure Partitioning.

6. Quick Sort
 Given again the array
40 20 10 80 60 50 7 30 100
0 1 2 3 4 5 6 7 8
7 20 10 30 40 50 80 60 100
<= Data [pivot] > Data [pivot]

7. Quick Sort
 Conquer
After dividing recursively sort each sub array
<= Data [pivot]
7 20 10 30
7 20 10 30
7 20 10 30
7 10 20 30
7 20 10 30
7 10 20 30
7 10 20 30

8. Quick Sort
 Conquer
After dividing recursively sort each sub array
 > Data [pivot]
50 60 80 100
50 80 60 100
50 80 60 100
50 80 60 100
50 60 80 100
50 80 60 100
50 60 80 100
50 60 80 100

9. Quick Sort
 Combine
 Once the conquer step recursively sorts, we are done. Why? All elements
to the left of the pivot, in array are less than or equal to the pivot and are
sorted, and all elements to the right of the pivot, in are greater than the
pivot and are sorted.
 The elements can't help but be sorted! So we combine the Left sub array +
Pivot + Right sub array.
 Output of the sorted array:
7 10 20 30 40 50 60 80 100

10. Quick Sort
 Quick sort Analysis
Scenarios Sorting Algorithm
Worst-Case O(n2)
Best-Case O(n log n)/O(n)
Average-CaseA O(n log n)

11. Quick Sort
Advantage of Quicksort:
 Efficient average case compared to any sort algorithm
 The elegant recursive definition
 The popularity due to its high efficiency

12. Quick Sort
Disadvantage of Quicksort
 The difficulty of implementing the partitioning algorithm
 The average efficiency for the worst case scenario, which is not offset by
the difficult implementation

13. Bubble Sort
Jimenez, Jake
Rojas, Davis Jacob
Miguel, Flynce

14. Bubble Sort
 The bubble sort makes multiple passes through a list. It compares adjacent items
and exchanges those that are out of order. Each pass through the list places the
next largest value in its proper place. In essence, each item “bubbles” up to the
location where it belongs.

15. Bubble Sort
 Bubble Sort is the simplest sorting algorithm that works by repeatedly swapping
the adjacent elements if they are in wrong order.
First Pass:
( 5 1 4 2 8 ) –> ( 1 5 4 2 8 )
 Here, algorithm compares the first two elements, and swaps since 5 > 1.
( 1 5 4 2 8 ) –> ( 1 4 5 2 8 )
 Swap since 5 > 4

16. Bubble Sort
( 1 4 5 2 8 ) –> ( 1 4 2 5 8 )
 Swap since 5 > 2
( 1 4 2 5 8 ) –> ( 1 4 2 5 8 )
 Now, since these elements are already in order (8 > 5), algorithm does not swap
them.

17. Bubble Sort
 Second Pass:
( 1 4 2 5 8 ) –> ( 1 4 2 5 8 )
( 1 4 2 5 8 ) –> ( 1 2 4 5 8 ), Swap since 4 > 2
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )

18. Bubble Sort
 Now, the array is already sorted, but our algorithm does not know if it is
completed. The algorithm needs one whole pass without any swap to know it is
sorted.
Third Pass:
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )

19. Bubble Sort
 Now, the array is already sorted, but our algorithm does not know if it is
completed. The algorithm needs one whole pass without any swap to know it is
sorted.
Third Pass:
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )

20. Bubble Sort
Advantages of Bubble sort:
 Easy to understand.
 Easy to implement.
 In-place, no external memory is needed.
 Performs greatly when the array is almost sorted.

21. Bubble Sort
Disadvantages Bubble sort:
 Very expensive, O(n2)O(n2)in worst case and average case.
 It does more element assignments than its counterpart, insertion sort.