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Introduction to Sorting

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By Saad Malik

Sorting: an operation that segregates items into groups according to specified criterion.

A = { 3 1 6 2 1 3 4 5 9 0 }

A = { 0 1 1 2 3 3 4 5 6 9 }

Consider:

Sorting Books in Library (Dewey system)

Sorting Individuals by Height (Feet and Inches)

Sorting Movies in Blockbuster (Alphabetical)

Sorting Numbers (Sequential)

There are many, many different types of sorting algorithms, but the primary ones are:

- Bubble Sort
- Selection Sort
- Insertion Sort
- Merge Sort
- Shell Sort
- Heap Sort

- Quick Sort
- Radix Sort
- Swap Sort

Most of the primary sorting algorithms run on different space and time complexity.

Time Complexity is defined to be the time the computer takes to run a program (or algorithm in our case).

Space complexity is defined to be the amount of memory the computer needs to run a program.

Complexity in general, measures the algorithms efficiency in internal factors such as the time needed to run an algorithm.

External Factors (not related to complexity):

Size of the input of the algorithm

Speed of the Computer

Quality of the Compiler

- An algorithm or function T(n) is O(f(n)) whenever T(n)'s rate of growth is less than or equal to f(n)'s rate.
- An algorithm or function T(n) is Ω(f(n)) whenever T(n)'s rate of growth is greater than or equal to f(n)'s rate.
- An algorithm or function T(n) is Θ(f(n)) if and only if the rate of growth of T(n) is equal to f(n).

Time complexity

Example

constant

O(1)

Adding to the front of a linked list

N

log

O(log

)

Finding an entry in a sorted array

N

linear

O(

)

Finding an entry in an unsorted array

N

N

n-log-n

O(

log

)

Sorting n items by ‘divide-and-conquer’

2

N

quadratic

O(

)

Shortest path between two nodes in a graph

3

N

cubic

O(

)

Simultaneous linear equations

1

1

5

5

8

8

(Binary) Finding 8:

(Linear) Finding 8:

9

9

21

21

22

22

50

50

Front

Initial:

1

0

Final:

6

3

http://www.cs.sjsu.edu/faculty/lee/cs146/23FL3Complexity.ppt

- Bubble Sort = n²
- Selection Sort = n²
- Insertion Sort = n²
- Merge Sort = n log(n)
- Quick Sort = n log(n)

- How do we improve the time efficiency of a program?
- The 90/10 Rule
90% of the execution time of a program is spent in

executing 10% of the code

- So, how do we locate the critical 10%?
- software metrics tools
- global counters to locate bottlenecks (loop executions,
function calls)

Possibilities (some better than others!)

- Move code out of loops that does not belong there
(just good programming!)

- Remove any unnecessary I/O operations (I/O operations
are expensive time-wise)

- Code so that the compiled code is more efficient
Moral - Choose the most appropriate algorithm(s) BEFORE

program implementation

Ann

98

Ann

98

Bob

90

Joe

98

Dan

75

Bob

90

Joe

98

Sam

90

Pat

86

Pat

86

Sam

90

Zöe

86

Zöe

86

Dan

75

original array

stably sorted

- A stable sort keeps equal elements in the same order
- This may matter when you are sorting data according to some characteristic
- Example: sorting students by test scores

www.cis.upenn.edu/~matuszek/cit594-2002/ Slides/searching.ppt

Ann

98

Joe

98

Bob

90

Ann

98

Dan

75

Bob

90

Joe

98

Sam

90

Pat

86

Zöe

86

Sam

90

Pat

86

Zöe

86

Dan

75

original array

unstably sorted

- An unstable sort may or may not keep equal elements in the same order
- Stability is usually not important, but sometimes it is important

www.cis.upenn.edu/~matuszek/cit594-2002/ Slides/searching.ppt

Step:

- 1. select the smallest element
- among data[i]~ data[data.length-1];
- 2. swap it with data[i];
- 3. if not finishing, repeat 1&2

20

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rio.ecs.umass.edu/ece242/slides/lect-sorting.ppt

- Place ith item in proper position:
- temp = data[i]
- shift those elements data[j] which greater than temp to right by one position
- place temp in its proper position

rio.ecs.umass.edu/ece242/slides/lect-sorting.ppt

temp

8

20

8

5

10

7

i = 1, first iteration

8

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10

7

---

8

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5

10

7

rio.ecs.umass.edu/ece242/slides/lect-sorting.ppt

temp

5

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10

7

i = 2, second iteration

5

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---

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rio.ecs.umass.edu/ece242/slides/lect-sorting.ppt

temp

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i = 3, third iteration

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---

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rio.ecs.umass.edu/ece242/slides/lect-sorting.ppt

temp

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i = 4, forth iteration

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---

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rio.ecs.umass.edu/ece242/slides/lect-sorting.ppt

http://www.cs.ubc.ca/spider/harrison/Java/sorting-demo.html