Quick Sort

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Quick Sort

• By HMA

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RECAP Divide and Conquer Algorithms

• This term refers to recursive problem-solving
  strategies in which 2 cases are identified
• A case for a direct (non recursive) solution
• A case for a recursive solution containing the
  following elements
• Dividing the input into smaller problems
• Finding recursive solutions for smaller problems
• Combining the solutions for the smaller problems

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Divide and Conquer Algorithms
Suppose we want to add the elements in an array
A1N
Add(A array, min, max)

1. If min max then return Amin
2. If min gt max then return 0
3. mid ? floor((minmax)/ 2)
4. FirstHalf ? Add(A,min, mid - 1)
5. SecondHalf ? Add(A, mid 1,max)
6. Return FirstHalf SecondHalf Amid

Complexity O(Dir) O(Div) O(Smaller)
O(Combining)
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Quicksort

• An element of the array is chosen. We call it the
  pivot element.
• The array is rearranged such that
• - all the elements smaller than the
  pivot are moved before it
• - all the elements larger than the
  pivot are moved after it
• Then Quicksort is called recursively for these
  two parts.

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Example
A 3 8 5 2 7 1 6 4
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Quicksort - Algorithm

• Quicksort (AL..R)
• if L lt R then
• pivot Partition( AL..R)
• Quicksort (A1..pivot-1)
• Quicksort (Apivot1...R)

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Partition Algorithm

• Partition (AL..R)
• p ? AL i ? L j ? R 1
• while (i lt j) do
• repeat i ? i 1 until Ai p
• repeat j ? j ? 1 until Aj p
• if (i lt j) then swap(Ai, Aj)
• swap(Aj,AL)
• return j

• Questions
• Why not choose for p the middle position in the
  array?
• Complexity?

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Example (again)
A 3 8 5 2 7 1 6 4
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Complexity Analysis

• Best Case
• Worst Case
• Average case

every partition splits half of the array
T(n) n 2T(n/2) T(1) 1 O(n log2n)
one array is empty one has all elements
T(n) n T(n-1) T(1) 1
O(n2 )
O(n log2n)