Spring 2016 Algorithms Midterm Exam (Show your work to get full credit!)

Transcription

1 Spring 2016 Algorithms Midterm Exam (Show your work to get full credit!) March 18, 2016 "Plagiarism is the intentional or unintentional use of the words or ideas of another without acknowledging their source." (Office of General Counsel, University of Akron) All the work shown on this exam must be your own work. All violations that occur during this exam will be reported to the Student Disciplinary Office as a violation of the Student Honor Code. The honor pledge statement: I do hereby resolve to uphold the Honor Code of the University of Akron. I shall refrain from all forms of academic dishonesty. By signing below, I pledge that I have neither given nor received help on this exam, nor am I aware of any infraction of the University of Akron s honor code. Student signature: Name Score 1) Solve the following recurrence: T(n) = 3T(n/3) + 18n a) using substitution method and express the result in O-notation. b) using recursion tree method and express the result in O-notation.

2 2) The golden ratio search is a divide-and-conquer technique for searching a key in a sorted array a. It works similar to the famous binary search but it narrows down possible locations with the aid of the golden ratio, i.e. instead of comparing with the middle element, Golden ratio search examines the element at index = 0.618*n, where n is the number of elements in the sorted array. For example, if you have an array that has 1000 elements, the algorithm will compare the key with the 618 th element instead of the 500 th element as you would do if you use binary search. If the key is less than a[index], the search continues with the first part of the array using the same strategy, if the key is bigger than a[index], the search continues with the second part of the array using same strategy, if the key = a[index], it stops and return the index. The process also stops when the array is empty, indicating the key is not in the array. a) Let T(n) be the worse-case time complexity of Golden ratio search. Give a recurrence for the worst-case running time. b) Solve the recurrence. c) Comparing with the regular binary search, give one possible advantage of binary search and one possible advantage of this algorithm?

3 3) RSA & Digital signature Assuming your public key is (e,n) and private key is (d, n). You are going to sign a message from a file. And assuming the hash code you obtained using SHA256 for the message is h, i.e. h=sha256(message). 1. Clearly list the steps you need to do to sign your message? 2. What steps your friend should follow to verify the authenticity of the message you send to her?

4 4) Consider a file containing the following text data: AAABBBAAB 1. Trace the LZW encoding process for the file in the same way done in class. Each "step" should produce a single codeword. Assume that the extended ASCII set will use codewords For each step in the encoding, be sure to show all of the information. Note: The ASCII value for 'A' is 65. output code (decimal) representing Codeword (decimal) representing 2. What s the size of your original file (saved using extended ASCII codes, 8bits/code). If each code in LZW uses 12bits, what s the size of the compressed file? 3. Assume that you have two variations of LZW, one that uses 12bit codewords and one that uses 16bit codewords. Explain how the performances of these variations compare and when (if at all) each would be preferred.

5 5) Consider the following graph: Use Dijkstra s algorithm to find the shortest path from vertex A to vertex F. Show your work.

6 6) (Convex Hull) Consider the following points S: a) Find the convex hull for S using Graham scan algorithm. Label your points and show your work! b) How many pops and pushes have you done to find the convex hull in part a)? c) (Graduate students) What s the time complexity of the algorithm? Why? Clearly justify your answer.

7 7) Professor Combine has proposed a new convex hull algorithm that combines the divide and conquer algorithm and the Graham Scan algorithm. The pseudo code of the algorithm is presented below: CombinedConvexHull(S) // finding the convex hull for the set of points (S) 1 if S <9 // the set has less than 9 points 2 return Graham_Scan(S) //return convex hull of S constructed using Graham scan 3 else //divide and conquer approach 4 y m = median(s) // return the median of y-coords of points in S. It can be done in O( S ) 5 S1 = lessm(y m,s) // find all the points whose y-coords y m 6 S2 = greaterm(y m,s) // find all the points whose y-coords > y m 7 CH1=CombinedConvexHull (S1) // convex hull of S1 8 CH2=CombinedConvexHull (S2) // convex hull of S2 9 return Combine(CH1, CH2) // merges the 2 convex hulls. // assume the complexity of merge is O( CH1 + CH2 ). CH1 =# of points on CH1 a. Give a recurrence for the worst-case running time of CombinedConvexHull for a set of n points. b. Solve the recurrence obtained in (a) using the master theorem. c. Give one reason you think Professor Combine s algorithm might be better than the standard divide-conquer convex hull? Justify your answer.