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HOMEWORK 9 Solution
Be sure to include your name at the beginning of each le! Assignment 9 include a programming portion and a written part. The programming portion must compile and consist of a single le ( hw09.cpp). The written portion should consist of a single le (hw09written) in a .pdf format. Be sure to include your name at the beginning of each le! You must hand in the le via NYU Classes.
You must hand in both les via NYU Classes.
Programming Part
Add the following methods to the BinarySearchTree class.
Implement the method contains recursively.
Implement a method that takes two keys, low and high, and prints all the objects X that are in the range speci ed by low and high. (i.e. all keys in the range [low, high].) Your program should run in O(k + h) time, where k is the number of keys printed and h is the height of the tree. Thus if k is small, you should be examining only a small part of the tree. Use a hidden recursive method and do not use an inorder traversal. Bound the running time of your algorithm using Big-Oh notation.
Ever feel contrarian? So much e ort is spent creating a balanced BST. In this problem1 you are to write a method called stringy that creates an unbalanced BST, a "stringy" BST, from a regular BST. If your tree contained
d
\
f
/ \ /
a c e
after running stringy, all nodes in your tree should have their left member variable contain the nullptr. Thus the tree should have height n 1.
5% extra credit will be given if you turn this assignment in on Nov 26, 2016 at 11:00 p.m.
This problem was suggested by Shahzaib.
1
In the example above, after calling stringy the tree would look like:
a
\
b
\
c
\
d
\
e
\
f
Your method must not create any new nodes2. I have removed the size method from the node class, so you do not need to update the size of each node.
Create a method called average_node_depth that computes the average depth of a node in the tree. e.g.
6
\
8
\
4
\
5
Then the average depth of a node is (0 + 1 + 2 + 2 + 3 + 1)=6 = 9=6 since there is one node at depth 1, two nodes at depth 2, one node at depth 3.3
Add a method to the binary search tree class that lists the nodes of a binary tree in level-order (It should rst list the root, then the nodes at depth 1, then the nodes at depth 2, and so on). Your algorithm should have a worst case running time of O(n), where n is the number of nodes in the tree. Explain how your algorithm ful lls this requirement. Hint: a Queue<Node * might be helpful in solving this problem.4
e.g. For the following tree:
31
\
12 53
/ \ / \
25 46 67
Printing the nodes of this tree in a level order would output: 31; 12; 53; 4; 25; 46; 67:
Please take the time to gure out how to solve this method on paper (using pictures) before you start coding.
3The average number of comparisons to nd a item in this tree is 9/6+1.
4Level-order traversal of a tree is also called breadth- rst traversal. I suggest you do not use recursion
2
(Extra Credit) You may do two of the following three extra extra credit problems. The points associated with each problem with be posted on Piazza.
Run empirical studies to estimate the average height of a node in a binary search tree by running 100 trial of inserting n random keys into an initially empty tree. For n = 210; 211; 212. For each n print the min depth of a node, the max depth of a node, and the average depth of a node.
Add5 bidirectional iterators to the binary search tree class. To do this:
add two extra pointers to the node class. Use these pointers to link each node to the next smallest and next largest node.6
add a tail node which is not part of the binary search tree7. Adding this node helps to make the code simpler
add the methods begin( ) and end( ) to the binary search tree class
See8 Shah’s Memory.pdf.
This problem was suggested by Shahzaib.
if you wish you can do this di erently. We might ask you to explain your code.
7You can add a head node - but it is not needed. 8This problem is from Shahzaib.
3
Written Part
Write the pseudo code for:
programming problem 1. For each method, write the 3 to 8 steps needed to solve this problem.
For the programming problems in question 2, determine the running times of your implementations using Big-Oh notation.
This code is modi ed from the code we discussed in class. Does this code perform correctly? If not, describe all problems of this code and x the code.
template <class Comparable
BinaryNode<Comparable * BinarySearchTree<Comparable::findMin( Node * t ) const
{
while( t-left != NULL && t != NULL ) t = t-left;
return t;
}
This code is modi ed from the code we discussed in class. Does this code perform correctly? If not, describe all problems of this code and x the code.
void insert( const Comparable & x, Node * t )
{
if( t == nullptr )
t = new Node{ x, nullptr, nullptr, 1 };
else if( x < t-element )
insert( x, t-left );
else if( t-element < x )
insert( x, t-right );
else
// Duplicate; do nothing t-size++;
}
For each of these questions, only print the number stored in the node. You do not need to print r or b which represent the color of the node.
Add 50, then 44, to this red-black tree (i.e. rst insert 50 and perform any operations necessary (using only the algorithm presented in class) so the tree is again a red-black tree, then insert 44 and perform any necessary operations (using only the algorithm presented in class) so the tree is again a red-black tree.)
Show the following:
Show the tree after inserting a value. If a violation occurred, state what the violation was (i.e. case 1, 2 or 3).
If the tree is not a red-black tree, use the algorithm we used in class to turn it into a red-black tree. Show each step.
4
22 b
17 b
30 b
14 r
18 r
Do not turn in. Please do this on your own. I will expect you to know how to do this for the exam. You can check your solution at https://www.cs.usfca.edu/ galles/visualization/RedBlack.html. Show the result of inserting (15, 4, 2, 8, 16, 9, 5, 10, 17, 18) into an initially empty red-black tree as described in class. Include the color of each node. Show the tree before and after each violation, and state which violation has occurred (i.e. case 1, case 2, or case 3).
Write the code needed to perform the method rightRotateRecolor for the RedBlackNode class. The method has the following prototype:
template <class Comparable
void RedBlackTree<Comparable::rightRotateRecolor( Node * & k2 )
When inserting an item into a Red-Black tree, what is the maximum number of pointer changes needed to adjust the tree? Explain your answer.
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