Sorting And Searching

1
Sorting And Searching

• CSE116A,B

2
Introduction

• The problem of sorting a 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.
• Sorting is a fascinating operation that provides
  a model for investigating many problems in
  Computer Science. Ex analysis and comparison of
  algorithms, divide and conquer, space and time
  tradeoff, recursion etc.

3
Sorting

• Comparison based selection, insertion sort (with
  online animations)
• Divide and conquer merge sort, quick sort (with
  animations from supplements of your text book)
• Priority Queue /Heap based heap sort
• Assume Ascending order for all our discussion

4
Selection Sort

• Select the first smallest element, place it in
  first location (by exchanging contents of
  locations), find the next smallest, place it in
  the next location, and so on.
• We will look at an example, an algorithm, analyze
  the algorithm, and look at code implementation.

5
Selection Sort Example
2
8
16
10
6
4
2
4
16
10
6
8
2
4
16
10
6
8
Sorted array
2
4
16
8
6
10
2
4
10
8
6
16
6
Selection Sort Pseudo Code

• Let cursor be pointer to first element of an n
  element list to be sorted.
• Repeat until cursor points to last but one
  element.
• 2.1 Search for the smallest element in the list
  starting from the cursor. Let it be at location
  target.
• 2.2 Exchange elements at cursor and target.
• 2.3 Update cursor to point to next element.

7
Sort Analysis

• Let el be the list n be the number of elements
  cursor 0 target 0
• while (cursor lt n-1)
• 2.1.1 target cursor
• 2.1.2 for (i cursor1 i lt n i)
• if (eli lt eltarget) target
  i
• 2.2 exchange (eltarget, elcursor) //
  takes 3 assignments
• 2.3 cursor
• (n-1) 4 2 (n (n-1) (n-2) ..1)
• 4n 4 2n(n1)/2 4n 4 n2 n n2 5n -
  4
• An2 B n C where A 1, B 5, C -4 for
  large n, drop the constant, lower order term in
  n, and the multiplicative constant to get big-O
  notation
• O(n2) quadratic sort

8
Insertion Sort
Unsorted array
10
Trivially sorted
8
10
Insert 8
6
8
10
Insert 6
2
6
10
8
Insert 2
2
6
16
10
8
Insert 4
Insert 16
9
Insertion Sort Pseudo Code

• Single element is trivially sorted start with
  first element
• Repeat for second to nth element of the list
• 2.1 cursor next location
• 2.2 Find a location to insert for listcursor
  by comparing
• and shifting
• let the location be target
• 2.3 Insert listtarget listcursor

10
Insertion Sort Analysis

• cursor 0
• while (cursor lt n)
• 2.1 cursor cursor 1
• 2.2 temp listcursor // save element to
  inserted
• 2.2.2 j cursor //find location
• 2.2.3 while (j gt 0 listj-1 gt temp )
• listj listj-1 //shift
  right
• j j 1
• // assert location found or hit left
  end(j0)
• 2.3 listj temp
• Worst case O(n2) quadratic
• Best case linear (when the list is already
  sorted)

11
Merge Sort

• Divide and Conquer
• Divide the list into two subsets s1, and s2
• (recurse) Sort s1 and s2 by divide and conquer
• (conquer) Merge the sorted s1 and s2.
• O(n log n) algorithm

12
Merge Sort (s)

• mergeSort(s)
• If S.size() gt 1
• 1. S1, S2 ? partition (S, n/2)
• 2. mergeSort(s1)
• 3. mergeSort(s2)
• 4. S ? merge(s1,s2)
• Lets look at examples.

13
Example
S1
S2
8
6
10
S11
S12
8
6
S121
S122