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Mergesort and Quicksort

Chapter 8

Kruse and Ryba

Sorting algorithms

• Insertion, selection and bubble sort have

quadratic worst-case performance

• The faster comparison based algorithm?

O(nlogn)

• Mergesort and Quicksort

Partitioning - Choice 1

• (^) First n-1 elements into set A, last element set B
• (^) Sort A using this partitioning scheme recursively
  • (^) B already sorted
• (^) Combine A and B using method Insert() (= insertion

into sorted array)

• (^) Leads to recursive version of InsertionSort()
  • (^) Number of comparisons: O(n^2 )
    • (^) Best case = n-
    • (^) Worst case = (^2)

2

^ 



n n

c i

n

i

Partitioning - Choice 2

• (^) Put element with largest key in B, remaining elements

in A

• (^) Sort A recursively
• (^) To combine sorted A and B, append B to sorted A
  • (^) Use Max() to find largest element  recursive SelectionSort()
  • (^) Use bubbling process to find and move largest element to right-most position  recursive BubbleSort()
• (^) All O(n^2 )

Example

• Partition into lists of size n/

[10, 4, 6, 3]

[10, 4, 6, 3, 8, 2, 5, 7]

[8, 2, 5, 7]

[10, 4] [6, 3]^ [8, 2]^ [5, 7]

[4] [10] [3][6] [2][8] [5][7]

Example Cont’d

• Merge

[3, 4, 6, 10]

[2, 3, 4, 5, 6, 7, 8, 10 ]

[2, 5, 7, 8]

[4, 10] [3, 6]^ [2, 8]^ [5, 7]

[4] [10] [3][6] [2][8] [5][7]

Merge Function

Evaluation

• Recurrence equation:

• Assume n is a power of 2

c 1 if n=

T(n) =

2T(n/2) + c 2 n if n>1, n=2k

Quicksort Algorithm

Given an array of n elements (e.g., integers):

• (^) If array only contains one element, return
• (^) Else
  • (^) pick one element to use as pivot.
  • (^) Partition elements into two sub-arrays:
    • (^) Elements less than or equal to pivot
    • (^) Elements greater than pivot
  • (^) Quicksort two sub-arrays
  • (^) Return results

Example

We are given array of n integers to sort:

Partitioning Array

Given a pivot, partition the elements of the array

such that the resulting array consists of:

1. One sub-array that contains elements >= pivot
2. Another sub-array that contains elements < pivot

The sub-arrays are stored in the original data array.

Partitioning loops through, swapping elements

below/above pivot.

pivot_index = 0 40 20 10 80 60 50 7 30 100

[0] [1] [2] [3] [4] [5] [6] [7] [8]

too_big_index (^) too_small_index

pivot_index = 0 40 20 10 80 60 50 7 30 100

[0] [1] [2] [3] [4] [5] [6] [7] [8]

too_big_index (^) too_small_index

1. While data[too_big_index] <= data[pivot] ++too_big_index

pivot_index = 0 40 20 10 80 60 50 7 30 100

[0] [1] [2] [3] [4] [5] [6] [7] [8]

too_big_index (^) too_small_index

1. While data[too_big_index] <= data[pivot] ++too_big_index