Sorting 2nd part

1
Tutorial 8

• Sorting (2nd part) Binary Search Tree

2
Merge Sort

• Key ideas
• Chop (unsorted) list into exactly half,
  recursively!
• Do it until size 1 (base case, by default 1
  item is sorted)
• When n is odd, the middle one can go to left or
  right sub list, just be consistent!
• When the recursion is winding up, do an efficient
  O(n) MERGING process.
• Simply compare front of sub list A and front of
  sub list B, the smaller is taken first!
• The overall complexity is O(n log n)
• At every recursion step we do O(n) merge process
• And we only do these merging process O(log2 n)
  times
• Compare with Tutorial 7 question 3.d part 2 (when
  g(n) O(n))
• Now, let us see an example in Q1!

3
Student Presentation

• Gr3
• Lim Wei Hong or Chia Jie Shen
• David Seo and Li Huan (or Tanvir Islam or Zhang
  Jianfei)
• Jacob Pang (or Hema Kumar or Robin Teh)
• Gr4
• Cynthia Tan or Sherilyn Ng
• Tan Peck Luan and Jasmine Choy
• Hanyenkno Afi or Wang Kang
• Overview of the questions
• Trace Merge Sort (1 student)
• Binary Search Tree (2 students)2a-b and 2c-d
• Convert a Binary Search Tree into Double Linked
  Circular List (1 student)

• Gr5
• Stephanie Teo
• Joyeeta Biswas
• Zhang Denan
• Gr6
• Laura Chua or Zhang Chao
• Brenda Koh and Gerard Lou
• Rasheilla or Chow Jian Ann

4
Q1 Trace Mergesort

• 4, 9, 2, 6, 1 3, 7, 8, 0, 5
• 4, 9, 2, 6, 1 3, 7, 8, 0, 5
• 4, 9, 2 6, 1 3, 7, 8 0, 5
• 4, 9 2 6 1 3, 7 8 0 5
• 4 9 1, 6 3 7 0, 5
• 4, 9 3, 7
• 2, 4, 9 3, 7, 8
• 1, 2, 4, 6, 9 0, 3, 5, 7, 8
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

5
Quick Sort

• Key ideas
• Partition (unsorted) list in O(n) steps around a
  reference number (pivot)
• Left sub list will be smaller than pivot, right
  sub list will be larger (or equal) than pivot
• Item equal to pivot can be placed on left or
  right sub list, just be consistent!
• After partitioning, pivot will definitely be in
  the correct place in sorted list.
• Choosing proper pivot is crucial for Quicksort!
• People usually take random pivot for better
  average performance
• Partitioning algorithm is the most complex part
  of quick sort.
• There are several partitioning algorithms out
  there, all in O(n)
• Example (as in lecture note) will be shown next
  week
• Then, recursively process the left and right sub
  lists in the same manner.
• Do it until size 1 (base case, by default 1
  item is sorted)
• It is on average O(n log n) too, if we use random
  pivot
• It can be faster than merge sort due to many
  reasons not discussed in CS1102
• Quick Sort will be discussed next week

6
Tree, Binary Tree, and BST

• Verify that you have understand these
• Basic concepts about Trees
• Extension of Linked List
• Node, Edges, Parent, Children, Root,Leaf,
  Internal Node, Level, Height, Size
• Basic concepts about Binary Trees
• Definition max 2 children (left and right),
  full, complete
• Implementation reference (linked) or array based
• Binary tree traversals Inorder, Preorder,
  Postorder, Level-order
• Basic concepts about Binary Search Trees
• Definition Binary Tree where BST property holds.
• Used in ADT Table (more advanced than List)
• List index (position) ? data
• Table key ? data

7
Binary Search Tree (BST)

• Binary Search Tree (assuming a roughly balanced
  tree)
• Insert O(height) gt O(log2 n)
• Start from root, at each step, determine whether
  to go to left or right
• Insertion will only occur at leaf!
• Search O(height) gt O(log2 n), similar to
  insertion
• Start from root, at each step, determine whether
  to go to left or right
• Stop when item is found or we reach the leaf but
  item not found
• Delete, 3 cases O(height) gt O(log2 n)
• Delete leaf (straightforward)
• Just delete that node
• Delete internal node with 1 child (either left or
  right child)
• Link the nodes only child with the nodes parent
• Delete internal node with 2 children
• Pick the inorder successor (or predecessor)to
  replace the content of the node to be deleted.
• Delete the actual copy of the replacement node
• Demo
• http//www.cs.jhu.edu/goodrich/dsa/trees/btree.ht
  ml

8
Q2 Trace BST Operations (1)

• From empty BST, insert 3, 6, 4, 10, 1, 2, 13, 8,
  7, 9, 15, 12, 11.
• Delete 1 and then 10
• (assumptioninorder successoris taken for
  deletionof node with twochildren)

9
Q2 Trace BST Operations (2)

• Traversals
• Inorder
• 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 15
• SORTED output! (Remember this)
• Preorder
• 3, 2, 6, 4, 11, 8, 7, 9, 13, 12, 15
• Postorder
• 2, 4, 7, 9, 8, 12, 15, 13, 11, 6, 3
• Complete Binary Tree?
• No, many blanks on left side of node 2!
• Delete 4 nodes (circled)

10
Q3 BST to a Sorted DLCL

• Give pseudocodeto convert a BSTinto a sorted
  DLCL!
• // Modify recursive inorder traversal!
• makeCircularDLL(root, doubleLinkedList)
• if(root.left ! NULL)
• makeCircularDLL(root.left, doubleLinkedList)
• doubleLinkedList.insertTail(root.item) //
  assume ADT linked list is ready
• if(root.right ! NULL)
• makeCircularDLL(root.right,
  doubleLinkedList)

11
Next Balanced BST Hashing

• If your BST is not balanced, its height can be as
  tall as O(n)!
• It degenerates into another Linked List
• This is not desirable!
• There are various proposals of Balanced BST
• AVL Tree (Adelson-Velski Landis)
• Rotate an unbalanced node during
  insertion/deletion
• This was inside last year CS1102 syllabus, but
  not this year.
• And many others Splay Tree, 2-3-4 Tree, Red
  Black Tree, etc
• Java has TreeMap ADT
• It is a balanced BST (Red-Black Tree)
• http//java.sun.com/j2se/1.5.0/docs/api/java/util/
  TreeMap.html
• The height of Balanced BST is expected to be
  O(log2 n)
• Next week something that probably faster than
  BST Hash Table

12
Extra Examples (1)

• Execute the following sequence of operations on
• An empty Binary Search Tree
• Note when you delete a node with two children,
  replace it with the inorder predecessor and
  delete the inorder predecessor.
• Show the final tree after the sequence of
  operations are executed.
• Insert(10) ? I(100) ? I(30) ? I(80) ? I(50),
• Delete(10),
• I(60) ? I(70) ? I(40),
• D(80),
• I(90) ? I(20),
• D(30),
• D(70)

13
Extra Examples (1) - Solution

• The Final BST
• If you did not manage to get this tree, re-do
  your solutions again!
• Your concept may be still incorrect.

14
Extra Examples (2-3)

• Note when you delete a node with two children,
  replace it with the inorder predecessor and
  delete the inorder predecessor.
• Example 2 Final BST
• I(1) ? I(2) ? I(3) ? I(4)
• D(3)
• I(3) ? I(5)
• D(4)
• Example 3 Final BST
• I(5) ? I(3) ? I(7) ? I(9)
• D(3)
• I(8) ? I(-2) ? I(1)
• D(5) ? D(9)
• I(-1) ? I(0)
• D(1)