COSC 2007 Data Structures II Final Exam. Part 1: multiple choice (1 mark each, total 30 marks, circle the correct answer)

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1 COSC 2007 Data Structures II Final Exam Thursday, April 13 th, 2006 This is a closed book and closed notes exam. There are total 3 parts. Please answer the questions in the provided space and use back of the paper, if needed. If you don t understand a question, or, if you don t remember a specific method, ask the instructor. Since this is an exam, you don t need to document your code; don t waste time for it. Student Name: GOOD LUCK Part 1: multiple choice (1 mark each, total 30 marks, circle the correct answer) 1) Most of Tree algorithms typically run in time O(d). What is d? A. The depth of the tree. B. The number of divisions at each level. C. The number of entries in each node. D. The number of nodes in the tree. E. The total number of entries in all the nodes of the tree. 2) Consider the node of a complete binary tree whose value is stored in data[i] for an array implementation. If this node has a left child, where will the right child's value be stored? A. data[i+1] B. data[i+2] C. data[2*i + 1] D. data[2*i + 2] 3) Suppose you have a list of names sorted in alphabetical order, already sorted in one of the data types below. The easiest way to print the names in reverse alphabetical order would be to use A. a BST B. a stack C. a queue D. a circular, single linked list 4) Depth-first search is best implemented using A. Recursion B. A queue C. A tree D. A graph 5) Breadth-first search is best implemented using A. a tree B. a stack C. a graph D. a queue 1

2 6) How are loops represented in an edge-list representation of a graph? A. A vertex will be on its own edge-list. B. The edge-list will be a circular linked list. C. The edge-list will be empty for that particular vertex. D. The edge-list will be full for that particular vertex. 7) A Java class contains only one data member of type byte, how many different instances (objects) of that class can there be in a given program at one time? A Java byte has a value in the range [0, 255]. a. 1 b. 255 c. 256 d. there is no limit to the number of instances 8) Suppose that we have implemented a priority queue by storing the items in a heap (using an array for the heap items). We are now executing a reheapification upward and the out-of-place node is at data[i] with priority given by data[i]. Which of the following boolean expressions is TRUE to indicate that the reheapification IS NOT YET DONE. A. (i > 0) B. (data[(i-1)/2] < data[i]) C. (i > 0) && (data[(i-1)/2] < data[i]) D. (i > 0) (data[(i-1)/2] < data[i]) 9) A simple graph has no loops. What other property must a simple graph have? A. It must be directed. B. It must be undirected. C. It must have at least one vertex. D. It must have no multiple edges. 10) What is the minimum number of nodes in a complete binary tree of height k? A. 2 k 1 B. 2 k C. 2 k 1 D. 2 k E. 2 k ) Suppose you have a directed graph representing all the flights that an airline flies. What algorithm might be used to find the best sequence of connections from one city to another? A. Breadth first search. B. Depth first search. C. A cycle-finding algorithm. D. A shortest-path algorithm. 12). What is the worse case time complexity for search in a general tree? A. log(n) B. n C. nlogn D. n 2 E. None of above 2

3 13). What is the worse case time complexity for search in a undirected connected graph? A. log(n) B. n C. nlogn D. n 2 E. None of above 14) If a 5 is inserted into the heap below, and the heap condition is restored using the method we discussed in lecture, into which position will the 5 go? i: A[i]: A. A[0] B. A[1] C. A[2] D. A[5] E. A[11] Suppose you start with an empty 2-3 tree, and then you add keys to it in the following order: 45, 30, 65, 70, 25, 80, 15, 40, 90, 75, 50, 60 15) What is the height of the resulting 2-3 tree? A. 2 B. 3 C. 4 D. 5 E. 6 16) What are there in the root of above 2-3 tree? A. 45 B. 40 C. 45, 70 D. 45, 65 E. 45, 80 If you began with an empty AVL tree, and then inserted the following keys into the tree in the following order: 40, 20, 15, 25, 30, 80, 75, 95, 90, 35, ) Which key would be in the root of the tree after inserting all the keys? A. 75 B. 40 C. 35 D. 30 E ) What is the height of the above AVL tree? A. 3 B. 4 C. 5 D. 6 19) A complete graph with 5 vertices has the following edges: A. 5 B. 6 C. 8 D. 10 E ) Which statement is correct? A. 2-3 tree is a balanced BST 3

4 B. 2-3 tree is a normal BST C. 2-3 tree is a AVL tree D. 2-3 tree is usually not a BST 21) Which statement is correct? A. AVL tree is a balanced BST B. AVL tree is a normal BST C. AVL tree is a 2-3 tree D. AVL tree is usually not a BST 22) What is the worse case time complexity for insertion in 2-3 tree? F. <logn G. log(n) H. n I. nlogn J. None of above 23) What is the worse case time complexity for insertion in AVL tree? A. <logn B. log(n) C. n D. nlogn E. None of above 24) What is the best definition of a collision in a hash table? A. Two entries are identical except for their keys. B. Two entries with different data have the exact same key. C. Two entries with different keys have the same exact hash value. D. Two entries with the exact same key have different hash values. 25) In an open-address hash table there is a difference between those spots which have never been used and those spots which have previously been used but no longer contain an item. Which function has a better implementation because of this difference? A. insert B. is_present C. remove E. size F. Two or more of the above functions 26) What kind of initialization needs to be done for an open-address hash table? A. None. B. The key at each array location must be initialized. C. The head pointer of each chain must be set to NULL. D. Both B and C must be carried out. 27) What is the expected number of operations needed to loop through all the edges terminating at a particular vertex given an adjacency matrix representation of the graph? (Assume n vertices are in the graph and m edges terminate at the desired node.) A. O(m) B. O(n) 4

5 C. O(m²) D. O(n²) 27) Which of the following statements is true? A. A graph can drawn on paper in only one way. B. Graph vertices may be linked in any manner. C. A graph must have at least one vertex. D. A graph must have at least one edge. 28) If you began with an empty AVL tree, and then inserted the following keys into the tree in the following order: 20, 40, 15, 25, 30, 80, 75, 95, 35, 90 Which key would be in the root of the tree after inserting all the keys? A. 75 B. 40 C. 35 D. 30 E ) If G is an directed graph with 20 vertices, how many boolean values will be needed to represent G using an adjacency matrix? A. 20 B. 40 C. 200 D ) How many linked lists are used to represent a graph with n nodes and m edges, when using an edge list representation, A. m B. n C. m + n D. m*n Part II Short Answer Questions (53 marks) Question 1: Which of the following are (5 marks): Complete Binary Trees: Full Binary Trees: Binary Search Trees: Heaps: 2-3 Trees: 5

6 (1) 10 (2) 10 (3) 10 / \ / \ / \ / \ / / \ / \ / / 1 (4) 10 (5) 10 (6) 10 / \ / \ / \ , / \ / \ / \ / \ / / \ (7) 10 (8) 10 / \ / \ / \ / / \ Question 2. Expression tree(6 marks total, 2 marks each) 2a) Draw this pre-fix expression as a binary expression tree: - * + A B C / D E 2b) Writing the equivalent post-fix expression. 2c) Convert the following post-order expression into a fully parenthesized expression. 8 4 / 2 * * + Question 3 Hashing (6 marks) 3a) Show the hash table that results from inserting the following keys, assuming the hash function h(k) = 3k 4 mod 11 and the linear probing style of collision handling. The elements associated with the keys are irrelevant; you must only show the keys: (2 marks) Keys:

7 b) Show the hash table that results from removing the following keys from the hashtable you finished with in question 3a: (2 marks) Keys: c) Show the hash table that results from inserting the following keys into the hashtable you finished with in question 8: (2 marks) Keys: Question 4: Answer true or false for each statement below (a-c). Using one sentence to justify your answer. (10 marks total, 2 marks each)., and answer the question for d and e. (a) To delete the element at the root of a binary search tree, we replace it by the rightmost leaf in the tree and then percolate down. (b) The order in which nodes are visited in preorder in a binary tree is exactly the reverse of the order in which nodes are visited in postorder. (c) Given a binary heap with n elements and an element x, one can determine whether x occurs in the heap in worst-case O(log n) time. (d) Give two different reasons to explain why the following binary tree is not a heap: 91 / \

8 / \ \ e) What is the minimum number of nodes in a heap of height h? What is the maximum number of nodes? Question 5) Binary Search Tree (10 marks) Sketch the binary tree left following each of the following deletion or deletions the starting point is always the tree shown above (that is, these deletions are not cumulative). Note: you may abbreviate the unaltered subtree as its subtree root node and the note unchanged. (5a) 2, then 4 (two items deleted show the tree after both items are gone) (5b) 9, taking as its replacement the in-order traversal predecessor 8

9 (5c) 9, taking as its replacement the in-order traversal successor. (5d) Insert the following two values into the BST above: 5, then 11 5e) Define the depth of a node in a tree and the height of a node in a tree. (While the terms apply to general trees, you may give your answer in terms of binary trees. Hint: the height of a BT is the height of the root node) Question 6) Given the following undirected weighted graph: (total 16 marks) B 3 A 1 7 C 6 5 D 2 E G 2 F 4 H 9

10 a) if a traversal were performed starting at vertex A, and visit are to be made in alphabetical order by vertex name where possible, in what order would vertices be visited during a: ( 6 marks) a1) depth-first search a2) breadth-first search b) Using Dijkstra s algorithm to find the shorted path from vertex A to vertex H. fill in one table below for each iteration of the algorithm until the result can be properly interpreted ( an iteration is made up of declaring a vertex to be done, followed by an examination of adjacent vertices). Fill the tables in from left to right, then top to bottom. You may not need to fill in all the tables to reach a solution. (7 marks) 0 F F F F F F F F 10

11 c) apply Prim s algorithm starting from vertex a to find the minimum spanning tree for the graph. Redraw the entire tree after each new edge-vertex pair is added. (3 marks) 11

12 Part III: Programming Questions (17 marks, 4 marks each, except last one which is 5) Question 1) A palindrome is a string that reads the same forwards and backwards. abba is a palindrome (also a Swedish rock group) while abab is not (but aba is). Write a method that is not associated with any class instance that takes a string S as input and returns true if S is a palindrome and false otherwise. Your method must use a data structure. You do not know the length of the string ahead of time. Note that a one letter string is a Palindrome. All the following questions will use the TreeNode class and BST class defined at the end of this test. Question 2) Write a method to list the keys that is, the contents of Node->Value in descending order (that is, reversed) to print one per line. It will be called with the root of the BST passed as the initial Node. Question 3) Write a recursive algorithm isbst(t) that takes only a reference (pointer) to the root of a binary tree and checks if the tree is a binary search tree. You may assume that the keys in the tree are all distinct. Beware of NULL pointers. You must handle them correctly. 12

13 Question 4) A full node is a node with 2 children. Give a recursive algorithm checkfullnode (T) that takes only a reference (pointer) to the root of a binary tree and computes the number of full nodes in the tree class TreeNode { Object item; Node left; Node right; } class BST { TreeNode head; boolean isempty(); void insert (KeyedItem, newitem); void delete (Comparable searchkey) throw TreeException; } 13