Sorting

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Sorting Recursion

• Briana B. Morrison
• Adapted from Alan Eugenio
• William J. Collins

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Topics

• Review
• Insertion Sort
• Selection Sort
• Bubble Sort
• This term
• Tree Sort
• Heap Sort
• Radix Sort
• New
• Merge Sort
• Quick Sort

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Selection Sort

• Loops through input and selects
  smallest/largest value and swaps with current
  location
• Worst case O(n2)
• Average case O(n2)

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Bubble Sort

• Loops through input doing adjacent comparisons
• Biggest/smallest element bubble to top/bottom
• Worst case O(n2)
• Average case O(n2)

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  FOR TREE SORT, EACH INSERTION
TAKES LOGARITHMIC TIME AT MOST, SO FOR
n INSERTIONS worstTime(n) IS O(n log n).
therefore averageTime(n) IS O(n log
n)    
 
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Heapify done.
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Sort done.
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Radix Sort

• Recall that Radix Sort uses a vector of queues to
  distribute and then collect from least
  significant position to most significant
  position.
• Performance is O(nd) where d is the length of
  the key ( of digits)
• Remember that Radix Sort IS NOT a comparison
  based sort.

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Divide Conquer Sorts

• Divide and conquer is a problem solving technique
  that uses recursion
• Divide problem into smaller problems that lead to
  recursive step or stopping condition
• Combine solutions to solve original
• Usually have 2 or more recursive calls that
  involve disjoint collections

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Mergesort vs. Quicksort

• Mergesort
• Divides collection to smallest solvable problem
• Combines solutions (this is where the work is)
• Quicksort
• Divides collection by a pivot (this is where the
  work is)
• Combines solutions

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The Merge Algorithm Example

• The merge algorithm takes a sequence of elements
  in a vector v having index range first, last).
  The sequence consists of two ordered sublists
  separated by an intermediate index, called mid.

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The Merge Algorithm (Cont)

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The Merge Algorithm (Cont)
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The Merge Algorithm (Cont)
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Partitioning Merging in mergeSort()
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Function calls in mergeSort()
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Quicksort Example

• The quicksort algorithm uses a series of
  recursive calls to partition a list into smaller
  and smaller sublists about a value called the
  pivot.
• Example Let v be a vector containing 10 integer
  values
• v 800, 150, 300, 650, 550, 500, 400, 350,
  450, 900
• Easy ways to choose the pivot
• First value
• Middle value
• Last value

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Quicksort Example

• Choose pivot as middle value, swap with first
  value.
• Example Let v be a vector containing 10 integer
  values
• v 800, 150, 300, 650, 550, 500, 400, 350,
  450, 900

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Quicksort Example (Cont)
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Quicksort Example (Cont)
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Quicksort Example (Cont)
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• Now swap the pivot with the value pointed to by
  scanDown
• Then generate the recursive calls

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Quicksort Example (Cont)
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Quicksort Example (Cont)
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Quicksort Example (Cont)
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Quicksort Example (Cont)
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Lets See Them Run

• // load the vectors with the same sequence of
  100000
• // random numbers in the range 0 to 999999
• for(i0i lt VECTORSIZEi)
• rndNum rnd.random(1000000)
• v1.push_back(rndNum)
• v2.push_back(rndNum)
• v3.push_back(rndNum)
• v4.push_back(rndNum)
• timeSort(v1,HEAPSORT,"Heap sort")
• timeSort(v2,MERGESORT,"Merge sort")
• timeSort(v3,QUICKSORT,"Quick sort")
• timeSort(v4,INSERTIONSORT,"Insertion sort")

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Sorting Animations

• http//www.cs.hope.edu/alganim/animator/Animator.
  html
• www2.dcc.ufmg.br/dorgival/applets/SortingPoints/S
  ortingPoints.html
• http//cg.scs.carleton.ca/morin/misc/sortalg/

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Summary Slide 1
- Divide-and-Conquer Algorithms - splits a
problem into subproblems and works on each
part separately - Two type of
divide-and-conquer strategy 1) mergesort
algorithm - Split the range of
elements to be sorted in half, sort each
half, and then merge the sorted sublists
together. - running time O(n
log2n), requires the use of an auxiliary
vector to perform the merge steps.
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Summary Slide 2
2) quicksort algorithm - uses a
partitioning strategy that finds the final
location of a pivot element within an
interval first,last). - The
pivot splits the interval into two parts, first,
pivotIndex), pivotIndex, last). All elements
in the lower interval have values ? pivot
and all elements in the upper interval have
values ? pivot. - running time
O(n log2n) - worst case of O(n2),
highly unlikely to occur