Sorting

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Sorting

• B.Ramamurthy

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Introduction

• The problem of sorting an initially unsorted
  collection of keys to produce an ordered
  collection is one of the richest in computer
  science.
• The richness derives from the fact that there are
  a number of ways of solving the sorting problem.
• It is a fascinating operation that provides a
  model for investigating many problem in Computer
  Science.

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Topics for Discussion

• Sorting Basics
• Priority queue methods
• Insertion based methods
• Divide and conquer methods
• Summary

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Comparison-based Sorting

• Most sorting methods decide how to rearrange the
  keys into order by first comparing the values of
  the keys.These methods are called
  comparison-based methods.
• Examples Selection sort, insertion sort, merge
  sort, quick sort, heap sort, shell sort and tree
  sort.
• The minimum average number of comparisons
  required to sort n keys using a comparison-based
  sorting method is proportional to n log n.

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Priority Queue Sorting Methods

• Using a tree heap and delete, repeat.
• Selection sort A realization of PQ method
  unsorted sub-array of keys represents the PQ.

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Selection Sort

• Number of comparisons (n-1)(n-2)2 1
  n(n-1)/2, or O(n )
• Number of exchanges O(n)
• Therefore it is a O(n ) algorithm.

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Heap Sort

• The PQ part of the method is represented by a
  heap.
• This implies that every time PQ part has be
  re-heaped when a key is moved. This is because
  the PQ may loose its heap property when the root
  is removed.

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Heap Sort Performance

• Heaping takes O(n)
• Remove key, move and sift up O(n log n)
• Since the latter is dominant for a large n,
• Heap sort is an O(n log n) algorithm.

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Insertion-based methods

• Main idea is to keep the keys in two list sorted
  and unsorted. Sorting involves removing an
  element from the unsorted list and placing it in
  the sorted list in the right order.
• Especially useful when the complete list is not
  available apriori.
• Insertion sort (O(n ) ) and tree sort (O (n log
  n)) are examples.

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Divide and Conquer methods

• Main idea
• 1. Divide the initially given unsorted array into
  two sub-arrays
• 2. Sort the two subarrays
• 3. Combine the two sorted subarrays into the
  overall solution.
• Merge sort and Quick sort are examples of this
  method. Lets look at these with a sample list.

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Example
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Performance of Merge and Quick sorts

• Merge sort is an O(n log n)
• Quick sort in average case is O(n log n) but is
  O(n ) in the worst case.

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