Assignment #1. Due March 8 (Sunday)
1. Draw the Binary Search Tree (BST) that results when you insert the keys E A S Y Q U e s T I O N,
in that order (associating the value i with the ith key, as per the convention in the text) into an
initially empty tree. Show all steps! How many compares are needed to build the tree?
2. Add to BST a method height() that computes the height of the tree. Develop two implementations:
a recursive method (which takes linear time and space proportional to the height), and a method
like size() that adds a field to each node in the tree (and takes linear space and constant time per
3. Give nonrecursive implementations of get() and put() for BST.
Partial solution : Here is an implementation of get():
public Value get(Key key)
Node x = root;
while (x != null)
int cmp = key.compareTo(x.key);
if (cmp == 0) return x.val;
else if (cmp < 0) x = x.left;
else if (cmp > 0) x = x.right;
The implementation of put() is more complicated because of the need to save a pointer to the parent node
to link in the new node at the bottom. Also, you need a separate pass to check whether the key is already
in the table because of the need to update the counts. Since there are many more searches than inserts in
performance-critical implementations, using this code for get() is justified; the corresponding change for
put() might not be noticed.
4. Prove that if a node in a BST has two children, its successor has no left child and its predecessor
has no right child.
5. Draw the sequence of BSTs that results when you delete the keys from the tree of Question 1, one
by one, in the order they were inserted.
6. Draw the 2-3 tree that results when you insert the keys E A S Y Q U e s T I O N in that order into an
initially empty tree.
7. Find an insertion order for the keys S E A R C H X M that leads to a 2-3 tree of height 1.
8. The figure below shows all the structurally different 2-3 trees with N keys, for N from 1 up to 6
(ignore the order of the subtrees). Draw all the structurally different trees for N = 7, 8, 9, and 10.
9. Prove that the height of a 2-3 tree with N keys is between ~⌊ log3 N⌋ (~0.63 log2 N (for a tree that
is all 3-nodes) and ~⌊log2 N⌋ (for a tree that is all 2-nodes).
10. Draw the red-black BST that results when you insert items with the keys E A S Y Q U e s T I O N in
that order into an initially empty tree. Show all steps!
11. Show the result of inserting n into the red-black BST drawn below (only the search path is shown,
and you need to include only these nodes in your answer). Show all steps!
12. Generate two random 16-node red-black BSTs. Draw them and then compare them with the
(unbalanced) BSTs built with the same keys.
13. Consider the B-Tree from page 20 of Lecture Notes 8. Insert first D and then L. Show all steps.
14. Demonstrate the insertion of the keys 5, 28, 19, 15, 20, 33, 12, 17, 10 into a hash table with
collisions resolved by chaining. Let the table have 9 slots, and let the hash function be
h(k) = k mod 9.
15. Professor Marley hypothesizes that substantial performance gains can be obtained if we modify
the chaining scheme so that each list is kept in sorted order. How does the professor’s
modification affect the running time for successful searches, unsuccessful searches, insertions,
16. Consider inserting the keys 10, 22, 31, 4, 15, 28, 17, 88, 59 into a hash table of length m = 11
using open addressing with auxiliary has function h’(k) = k mod m. Illustrate the result of
inserting these keys using linear probing, quadratic probing with c1 = 1 and c2 = 3, and using
double hashing with h2(k) = 1 + (k mod (m-1)).
17. Write a program to find values of a and M, with M as small as possible, such that the hash function
h(k) = (a k) mod M for transforming the kth letter of the alphabet into a table index produces distinct
values (no collisions) for the keys S E A R C H X M P L.
The result is known as a perfect hash function.
18. Consider the graph G presented below:
Provide its representation using:
a. Adjacency matrix
b. Array/list of edges
c. Adjacency lists
19. Prove Theorem from page 6 of Lecture Notes 10.
20. Add a method hasEdge() to Graph (page 526 of the textbook) which takes two int arguments v and
w and returns true if the graph has an edge v-w, false otherwise.
21. Consider the graph G below:
Find a Depth-First Search tree T for the above graph starting with the vertex 1. Show all the
vertices as they are discovered in sequence starting from 1 to the last vertex included in T, as
shown on page 17 of Lecture Notes 10.
22. Prove that every connected graph has a vertex whose removal (including all adjacent edges) will
not disconnect the graph, and write a DFS method that finds such a vertex.
Hint: Consider a vertex whose adjacent vertices are all marked.
23. We can implement Depth First Search without using recursion, by implementing stack explicitly.
The below pseudo-code describes such implementation:
Implement (and test) the above algorithm in Java. Consider two cases:
a. A graph is represented by adjacency list.
b. A graph is represented by adjacency matrix.
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