We don t have much time, so we don t teach them [students]; we acquaint them with things that they can learn. Charles E. Leiserson

Transcription

1 Homework 3 CS Data Structures Fall 2018 Dec 6, 2018 Name: Collaborators: We don t have much time, so we don t teach them [students]; we acquaint them with things that they can learn. Charles E. Leiserson This problem set covers material in Chapter 10, 12, 13, 22 of the textbook, Introduction to Algorithms, Third Edition (CLRS). To complete your homework, you may ONLY consult the following materials: Lecture slides posted on the class website. Course notes you or others took during lecture. The required text. Websites that may clarify the concepts covered in the material but do not in any way provide solutions to the problems. Problem Points Score Total: 92 Remember, your goal is to communicate. Full credit will be given only to correct solutions which are described clearly. Convoluted and obtuse answers will receive low marks. Collaboration on the problem sets is not only allowed, it s encouraged. It s proven that students who work on the homework in small groups perform better in the course than those that don t.

2 1. (11 points) Asymptotic Notation and Run-time Efficiency. (a) (3 points) If an algorithm s running time can be expressed as a function f(n) = Ω(n) + n 2, which one of the following asymptotic bounds is impossible for the running time? 1. Θ(n) 2. Θ(n 2 ) 3. Θ(n 3 ) 4. Θ(2 n ) (b) (3 points) Assume an algorithm runs in Θ(n 2 ), which one of the following is a wrong asymptotic notation for it? 1. O(n 3 ) 2. O(n 2 ) 3. Ω(n 2 ) 4. All of the above. (c) (5 points) Rearrange the following functions based on their growth rates, from the least to the greatest. In addition, underline those functions which have the same asymptotic bound. 2 n/2, log e n, n log 10 n, n!, 2 n, log 2 n n, n,

3 2. (19 points) Data Structures. (a) (3 points) Which one of the following statement is false for linked lists, assuming there is only one entry point to the list - head? 1. It takes constant time to insert an element at the front of a linked list. 2. It takes constant time to insert an element in the end of a linked list. 3. It takes linear time to insert an element in the middle of a linked list. 4. It takes linear time to remove an element in the end of a linked list. (b) (3 points) If we read a sequence of items < A, B, C, D, E, F > from a file and maintain some of the items in a stack by push and pop operations, which one of the following stack is possible, assuming that each item can be pushed into the stack only once. 1. top < D, E, F > bottom 2. top < F, C, A > bottom 3. top < C, D, E > bottom 4. top < E, D, F > bottom (c) (3 points) For hashing by open addressing, we need to decide the table size m and some parameters so that the table can be fully utilized. What does that actually mean? 1. Allocate a table with size m equal to the number of elements stored. 2. Decide m and other parameters so that for any element, the initial m probe positions are distinct. 3. Decide m and other parameters to reduce the probability of collisions. 4. Decide m and other parameters so that no collisions happen. 3

4 (d) (10 points) If a sequence of keys arrives in this order: < 47, 7, 39, 80, 23 >, successively insert each of them into the following tables, using all three of the open-addressing methods. Assume the primary hash function is h 1 (k) = k mod 8, the secondary hash function is { 1 + (k mod 7) if k mod 7 is even h 2 (k) = k mod 7 if k mod 7 is odd and the constants for quadratic probing are c 1 = c 2 = 1 2. index linear quadratic double

5 5

6 3. (20 points) Splay Trees. (a) (10 points) What are the sequence of Splay Tress after inserting each character in the list < a, j, i, b, c, h, g, d, e, f > to an initially empty Splay Tree? Be sure to draw one tree for each insertion. 6

7 (b) (10 points) Given the tree constructed in part (a), what are the sequence of trees after deleting each character in the list (i.e., delete a, delete j,...)? Be sure to draw one tree for each deletion. 7

8 4. (20 points) AVL Trees. (a) (10 points) What are the sequence of AVL trees after inserting each character in the list < a, j, i, b, c, h, g, d, e, f > to an initially empty AVL tree? Be sure to draw one tree for each insertion. 8

9 (b) (10 points) Given the tree constructed in part (a), what are the sequence of trees after deleting each character in the list (i.e., delete a, delete j,...)? Be sure to draw one tree for each deletion. 9

10 5. (6 points) Priority Queues. Write a pseudo-code procedure called Compare to decide the relative order of two events. The events are stored in a Max-Priority Queue, and the method is needed to compare two different events. The method should return 1 if a has a higher priority than b, and 1 otherwise. An event a has a higher priority than another event b if a s priority value > b s priority value. If both a and b have the same priority, then the event with earlier arrival time has a higher priority. Two events cannot arrive at the same time. Assume that priority and time are attributes of the events. Compare(a, b) // both a and b are events 6. (6 points) Tree Traversal. Given a binary tree, write a pseudo-code procedure to perform a level-order traversal to print all nodes in the tree. Level-order-print(x) // x is the root of the tree 10

11 7. (10 points) Graph Algorithms. Given the weighted and directed graph below: (a) (3 points) Find a discovering sequence of vertices in a BFS search if A is the source vertex. (b) (3 points) Find a discovering sequence of vertices in a DFS search if A is the source vertex. 11

12 (c) (4 points) Find a topological sequence of vertices. 12