CSSE 230
Data Structures and Algorithm Analysis

Written Assignment 7 - 80 points

To Be Turned In

Submit to the WA7 drop box on ANGEL. Late days may be used for written assignments.

1. Don’t forget: You should be keeping an individual log about your team project work so that you can write a performance evaluation for each of your teammates. You do not need to turn in this log, but I wanted to remind you while I had your attention.

2. (30 points) Consider the following directed graph:

   
   1. (5) Show the first three rows of an adjacency matrix for this graph. Label the rows and columns.
   2. (5) Show an adjacency list representation of this graph.
   3. (5) Show an edge list representation of this graph.
   4. (15) Which of the above representation will use the least space? Support your answer with calculations. Be explicit about your assumptions concerning details of the representations.
3. (15 points)  Prove by mathematical induction that n! ≥ 2ⁿ for all integers n≥4. (It can be proven other ways, but this is a good example for practicing induction, so that is what you should do).
4. (15 points) Weiss exercise 20.5 [20.5]. In this problem you will check your understanding of collision resolution techniques in hash tables.
5. (20 points) Weiss Exercise 21.3[21.3]. The add and buildHeap algorithms are on pages 815 [753] and 819[757], respectively. Do this one carefully. It will be hard to know how to give partial credit for incorrect answers.

Optional practice problem - not to be turned in.

(xx points) Start with the following Binary Search Tree:

See the hint at the end!

1. Is this an AVL tree? ____ If not, rearrange it so that it is height-balanced.
2. Draw the tree after insertion of a node containing 11, using the usual BST insertion algorithm.
3. Is the new tree AVL? ______ If not, name the node where the rotation should be done, according to the algorithm from class. ______ Single or double rotation? ____________ If you need to do a rotation, draw the resulting tree.
4. Delete the element 5 from the original tree (not the one from part c), using the BST deletion algorithm described in class and in the textbook. Draw the new tree.
5. Is the new tree AVL? ______ If not, name the node where the rotation should be done, according to the algorithm from class. ______ Single or double rotation? ____________ If you need to do a rotation, draw the resulting tree.
6. Is your new tree (if any) AVL? ______ If not, name the node where the rotation should be done, according to the algorithm from class. ______ Single or double rotation? ____________ If you need to do a rotation, draw the resulting tree.
7. Add the element 5 to the tree from part (f) using the BST algorithm. Draw the new tree.
8. Is the new tree AVL? ______ If not, name the node where the rotation should be done, according to the algorithm from class. ______ Single or double rotation? ____________ If you need to do a rotation, draw the resulting tree.
9. Add the element 12 to the tree from part (h). Draw the new tree.
10. Is the new tree AVL? ______ If not, name the node where the rotation should be done, according to the algorithm from class. ______ Single or double rotation? ____________ If you need to do a rotation, draw the resulting tree.
11. Add the element 7 to the tree from part (j). Draw the new tree.
12. Is the new tree AVL? ______ If not, name the node where the rotation should be done, according to the algorithm from class. ______ Single or double rotation? ____________ If you need to do a rotation, draw the resulting tree.

Hint: In my solution, for all the sub-parts together, I did three single rotations and no double rotations. The first four leaf nodes in my final tree were 3, 5, 7, and 9. (In my first attempt, I misread part d) and deleted from the tree in part c) rather than the original tree. With that mistake, for all the sub-parts together, I did two single rotations and one double rotation and the first four leaf nodes in my final tree were also 3, 5, 7, and 9.)