Sorting

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Sorting

Lecture 8

• Prof. Sin-Min Lee
• Department of Computer Science

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Preview for todays presentation

• What is sorting?
• Why do we sort?
• Three sorting methods
• Insertion Sort
• Selection Sort
• Bubble Sort
• Which method is better?

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What is sorting?

• It is a process of re-arranging a given set of
  objects in a specific order. (in ascending or
  descending order)
• Ex Objects are sorted in telephone books, in
  the tables of contents, in dictionaries, in
  income tax files, in libraries, in warehouses and
  almost everywhere that stored objects have to be
  searched and retrieved.

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Why do we sort?

• It is a relevant and essential activity in data
  processing.
• It is efficient to search in a ordered data set
  than an unordered data set.

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Sorting methods / algorithms
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Before proceeding . . .

• Terminology and Notation
• We are given items
• a1, a2, a3, . . . . . . , an
• then, we want to sort these items into an order
• ak1, ak2,ak3, . . . . . ., akn
• such that,
• f(ak1) ? f(ak2) ? f(ak3) ? . . . . . . ? f(akn)

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Judging the speed of sorting algorithms

• Best case - it occurs when the data are already
  in order.
• Average case - it occurs when the data are in
  random order.
• Worst case - it occurs when the data are in
  reverse order.

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Judging the speed sorting algorithm contin.

• Does the algorithm exhibit natural or unnatural
  behavior?
• Natural behavior exhibits if the sort works least
  when the list is already in order, works harder
  as the list becomes less ordered, and works
  hardest when a list is in inverse order.
• Unnatural behavior exhibits when minimum number
  of comparisons is needed when data are in reverse
  order and maximum if data are already in order.
• Does it rearrange data with equal keys?

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Introduction To Sorting

• Only concerned with internal sorting where all
  records can be sorted in main memory
• We are interested in analyzing sorting algorithms
  for worst-case performance and average performance

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Insertion Sort

• Initial order SQ SA C7 H8 DK
• Step 1 SQ SA C7 H8 DK
• Step 2 C7 SQ SA H8 DK
• Step 3 C7 DK H8 SQ SA

CClub DDiamond SSpade HHeart
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Insertion Sort Another Example
Original 34 8 64 51 32 21 Positions
Moved
After p2 8 34 64 51 32 21 1 After
p3 8 34 64 51 32 21 0 After p4 8
34 51 64 32 21 1 After p5 8 32 34
51 64 21 3 After p6 8 21 32 34 51
64 4
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Insertion Sort

• Step 1 sort the first two items of the array.
• Step 2 insert the third item into its sorted
• position in relation to the first two items.
• Step 3 the fourth element is then inserted into
  the list of three elements
• Continue until all items have been sorted

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Pseudocode - Insertion Sort

• for (i 1 i lt n i)
• place datai in its proper
• position
• move all elements dataj gt
• datai by one position

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Implementation of the Insertion Sort

• Templateltclass genTypegt
• void Insertion (genType data , int n)
• register int i,j
• genType temp
• for (i 1 i lt n i)
• temp datai
• j i
• while (j gt 0 temp lt dataj-1)
• dataj dataj-1
• j--
• dataj temp

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Example sort the array 5 2 3 8 1

• a0 a1 a2 a3 a4
• i 1 5 2 3 8 1
• j 1,0
• temp2
• i2 2 5 3 8 1
• j2
• temp3
• i3 2 3 5 8 1
• j1,3
• temp8

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Insertion sort example contin.1

• a0 a1 a2 a3 a4
• i4 2 3 5 8 1
• j4
• temp1
• i4 2 3 5 8
• j3
• temp1
• i4 2 3 5 8
• j2
• temp1

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Insertion sort example contin.2

• a0 a1 a2 a3 a4
• i4 2 3 5 8
• j1
• temp1
• i4 2 3 5 8
• j0
• temp1
• i5 1 2 3 5 8
• j0
• temp1

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Number of Exchange for each case

• Insertion Sort
• Best Case Cmin n-1
• O(n) Mmin 2(n-1)
• Average Case Cave 1/4(n2n-2)
• O(n2) Mave 1/4(n29n-10)
• Worst Case Cmax 1/2(n2n)-1
• O(2) Mmax 1/2(n23n-4)
• // C --gt Comparisons M --gt Moves //

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Insertion Sort Running Time

• The average case and worst case running times are
  quadratic.
• The best case running time is linear.
• The algorithms is quick when the starting array
  is nearly sorted

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Selection Sort

• Step1 Select the element with the lowest
• value and exchanges it with the
• 1st element
• Step2 From the remaining n-1 elements,
• the element with the next-lowest
• value is found and exchanged with
• the 2nd element.
• Continue up to the last two elements.

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Pseudocode - Selection Sort

• For(i 0 i lt n-1 i)
• select the smallest element among datai, . .
  . , datan-1
• swap it with datai

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Implementation of the Selection Sort

• Templateltclass genTypegt
• void Selection(genType data, int n)
• register int i, j, least
• for (i 0 i lt n-1 i)
• for (j i1, least i j lt n j)
• if (dataj lt dataleast) least
  j
• if (i ! least) Swap
  (dataleast, datai)

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Example sort the array5 2 3 8 1

• a0 a1 a2 a3 a4
• i0 5 2 3 8 1
• j1,2,3,4,5 ? 2 4 8 ? swap
• least 0,1 1 2 3 8 5
• i1 1 2 3 8 5
• j2,3,4,5 1 ? 8 5
• least 1 1 2 3 8 5

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Selection Sort example contin.

• a0 a1 a2 a3 a4
• i2 1 2 3 8 5
• j3,4,5 1 2 ? 8
• least2 1 2 3 8 5
• i3
• j4,5 1 2 3 8 5
• least3,4 1 2 3 ? ? Swap
• finally i4 1 2 3 5 8

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Selection Sort Example 2

• 44 55 12 42 94 18 06 67
• -------------------------------------------------
  ------------------
• 06 55 12 42 94 18 44 67
• -------------------------------------
• 06 12 55 42 94 18 44 67
• ---------------------------------
• 06 12 18 42 94 55 44 67
• 06 12 18 42 94 55 44 67
• ----------------------
• 06 12 18 42 44 55 94 67
• 06 12 18 42 44 55 94 67
• -----------
• 06 12 18 42 44 55 67 99

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Number of exchange for each case

• Selection Sort
• for all cases 1/2(n2 - n) comparisons
• Best Case Mmin 3(n - 1)
• Worst Case Mmax n2/4 3(n - 1)
• Average Case n(ln ny), where y is
• Eulers constant
• (about 0.577216)

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Selection Sort Running Time

• All cases running time for Selection Sort are
  quadratic.
• O(n2)