Sorting Chapter 9

Objectives Selection Sort, Insertion Sort, Quick Sort, and Merge Sort.

Introduction Common problem: sort a list of values, starting from lowest to highest. – Telephone directory – Words of dictionary in alphabetical order – Students names listed alphabetically Choose a criteria which is used to order data Given a list of records that have keys, we use these keys to define an ordering of the items in the list.

Example 1

The Insertion Sort Algorithm Given an array of integers The Insertion Sort algorithm views the array as having a sorted side and an unsorted side. The sorted side starts with just the first element, which is not necessarily the smallest element. The sorted side grows by taking the front element from the unsorted side.

Example 2

Example 3

Example 4

The Selection Sort Algorithm Start by finding the smallest element. Swap the smallest entry with the first element. Part of the array is sorted. Find the smallest element in the unsorted side. Swap with the front of the unsorted side. The size of the sorted side is increased by one element. Continue until the unsorted side has just one number. Why?

The Selection Sort Algorithm (cont.)

Example 1

Example 2

Quick sort 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 Quick sort two sub-arrays Return results

Pick 6 5 3 1 8 7 2 4 2 5 3 1 8 7 6 4 2 1 3 5 8 7 6 4 1 2 3 5 8 7 6 4 1 2 3 4 8 7 6 5 1 2 3 4 5 6 7 8

Total comparisons Assume that keys are random, uniformly distributed. Best case running time: O(n log2n) Worst case running time: O(n2)!!!

Quick sort: Worst Case

Partitioning Array Given a pivot, partition the elements of the array such that the resulting array consists of: One sub-array that contains elements >= pivot 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.

Example 1 40 20 10 80 60 50 7 30 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] 40 20 10 80 60 50 7 30 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] 40 20 10 80 60 50 7 30 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] 40 20 10 80 60 50 7 30 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index 40 20 10 80 60 50 7 30 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index 40 20 10 80 60 50 7 30 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] 40 20 10 80 60 50 7 30 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] 40 20 10 30 60 50 7 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. 40 20 10 30 60 50 7 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. 40 20 10 30 60 50 7 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. 40 20 10 30 60 50 7 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. 40 20 10 30 60 50 7 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. 40 20 10 30 60 50 7 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. 40 20 10 30 60 50 7 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. 40 20 10 30 7 50 60 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. 40 20 10 30 7 50 60 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. 40 20 10 30 7 50 60 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. 40 20 10 30 7 50 60 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. 40 20 10 30 7 50 60 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. 40 20 10 30 7 50 60 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. 40 20 10 30 7 50 60 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. 40 20 10 30 7 50 60 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. 40 20 10 30 7 50 60 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. Swap data[too_small_index] and data[pivot_index] 40 20 10 30 7 50 60 80 100 pivot_index = 0 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

While data[too_big_index] <= data[pivot] While data[too_small_index] > data[pivot] --too_small_index If too_big_index < too_small_index swap data[too_big_index] and data[too_small_index] While too_small_index > too_big_index, go to 1. Swap data[too_small_index] and data[pivot_index] 7 20 10 30 40 50 60 80 100 pivot_index = 4 [0] [1] [2] [3] [4] [5] [6] [7] [8] too_big_index too_small_index

Partition Result 7 20 10 30 40 50 60 80 100 [0] [1] [2] [3] [4] [5] [6] [7] [8] <= data[pivot] > data[pivot]

Recursion: Quicksort Sub-arrays 7 20 10 30 40 50 60 80 100 [0] [1] [2] [3] [4] [5] [6] [7] [8] <= data[pivot] > data[pivot]

Merge Sort Problem: Given n elements, sort elements into non-decreasing order Apply divide-and-conquer to sorting problem If n=1 terminate (every one-element list is already sorted) If n>1, partition elements into two sub-arrays; sort each; combine into a single sorted array How do we partition?

Example 1 Partition into lists of size n/2 [10, 4, 6, 3, 8, 2, 5, 7] [10, 4, 6, 3] [8, 2, 5, 7] [10, 4] [6, 3] [8, 2] [5, 7] [4] [10] [3][6] [2][8] [5][7]

Example Cont’d Merge [2, 3, 4, 5, 6, 7, 8, 10 ] [3, 4, 6, 10] [2, 5, 7, 8] [4, 10] [3, 6] [2, 8] [5, 7] [4] [10] [3][6] [2][8] [5][7]

Example 2 6 5 3 1 8 7 2 4 6 5 3 1 8 7 2 4 6 5 3 1 8 7 2 4 5 6 1 3 7 8 2 4 1 3 5 6 2 4 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8