CSE 247 Data Structures and Algorithms Fall Exam I

Transcription

1 CSE 247 Data Structures and Algorithms Fall 2016 Given: 29 February 2016 Exam I Due: End of session This exam is closed-book, closed-notes. No electronic devices or resources of any kind are allowed. The exception is the cheat sheet as described on our course web pages on which you may have notes to consult during the exam. Your workmustbelegible. Work that isdifficulttoread willreceive nocredit. Donot dwell over punctuation or exact syntax in code; however, be sure to indent your code to show its structure. You must sign the pledge below for your exam to count. Any cheating will cause the students involved to receive an F for this course. Other action may be taken. Your work must be completed on these pages. There are blank pages at the of the exam if you need them. Don t sp too much time staying stuck on a question: move on and return to the question later. You must fill in your identifying information correctly. When you reach this point in the instructions, please give the professor or TA a meaningful glance. Name (print clearly): Print clearly the following information: Student 6-digit ID (print really clearly): Problem Possible Number Points Total 100 Received Points Pledge: On my honor, I have neither given nor received any unauthorized aid on this exam. Signed: (Be sure you filled in your information in the box above!)

2 CSE 247 (Exam I) 2 Due by End of session 1. (20 points) (a) (c) (15 points) (a) n 2 = O(g(n)) for which of the following definitions of g(n)? Circle all that apply: g(n) = 1 g(n) = n g(n) = nlogn g(n) = n 2 n g(n) = n 2 logn g(n) = n 3 (b) n 2 = Ω(g(n)) for which of the following definitions of g(n)? Circle all that apply: g(n) = 1 g(n) = n g(n) = nlogn g(n) = n 2 n g(n) = n 2 logn g(n) = n 3 (c) n 2 = Θ(g(n)) for which of the following definitions of g(n)? Circle all that apply: g(n) = 1 g(n) = n g(n) = nlogn g(n) = n 2 n g(n) = n 2 logn g(n) = n 3 (d) (5 points) For just one of the items you circled above, provide a detailed proof below concerning n 2 and its stated relationship to g(n).

3 CSE 247 (Exam I) 3 Due by End of session 2. (20 points) (a) (2 points) True or false? 2 n = O(2 n+1 ) (b) (5 points) Provide below a proof of your claim: (c) (2 points) True or false? 2 n = Ω(2 n+1 ) (d) (6 points) Restate 2 n = Ω(2 n+1 ) in terms of big-o: = O( ) (e) (5 points) Provide below a proof of your true/false claim for problem 2c:

4 CSE 247 (Exam I) 4 Due by End of session 3. (20 points) Recall that the median value of a set is the value that separates the higher half of the set s values from the set s lower values. For example, given the set the median value is 240. {247,131,132,332,240} Consider a set S of arbitrary(and distinct) integer values(not necessarily the set shown above). Let n denote the size of set S, and assume throughout this problem that n is odd. All of S s elements have already been inserted into a MinHeap named heap. (a) (8 points) Below, write the method int findmedian(minheap heap, int n) that returns the median value from set S, assuming S s (and therefore heap s) size is the odd integer n. You will receive full credit only if your method runs in O(nlogn) time. You are free to use any methods in the API for MinHeap: insert(int), int extractmin(), and boolean isempty(). Also, it doesn t matter what the heap looks like when a call to your findmedian method is done. Continued on next page...

5 CSE 247 (Exam I) 5 Due by End of session (b) (2 points) Explain below why your solution runs in O(nlogn) time: (c) (10 points) Now suppose that we require findmedian to ensure that the heap contains the same elements before and after its call. It is not required that any element be in the same heap location. In observance of that requirement, when x S is extracted by your findmedian method, you will add it to a suitable data structure. Before returning, findmedian will call heap.insert(x) for each element x held in that data structure. i. (2 points) While elements of the heap may change position after calling your findmedian(minheap heap), one particular element is guaranteed to be in the same position. A. (1 points) Which element is guaranteed to be in the same position? B. (1 points) In terms of the array used inside MinHeap, at which index is the position of that element? ii. (1 points) Pick one of the following data structures to save the extracted elements of S (circle exactly one): Stack Queue Continued on next page...

6 CSE 247 (Exam I) 6 Due by End of session iii. (7 points) Below, show how you would use your circled data structure in findmedian to ensure that heap contains all elements of S on return. Do this by providing a complete findmedian below that incorporates your chosen data structure. Be sure to reference the specific methods of your data structure and explain how they would be used.

7 CSE 247 (Exam I) 7 Due by End of session 4. (20 points) For each of the program fragments below, let T(n) represent, worst-case, how many times statement s executes in terms of n, an input value to each fragment. (a) Fragment: for i=1 to n for j=1 to n s Worst-case, statement s above executes O( ) and Ω( ) times. (b) Fragment: for i=1 to n for j=1 to 1000 s Worst-case, statement s above executes O( ) and Ω( ) times. (c) Fragment: for i=1 to n for j=1 to i s Worst-case, statement s above executes O( ) and Ω( ) times. (d) Fragment: for i=1 to n if Math.random() < // very unlikely this is true for j=1 to n s Worst-case, statement s above executes O( ) and Ω( ) times.

8 CSE 247 (Exam I) 8 Due by End of session 5. (10 points) (a) (5 points) Consider a MinHeap h of n integers, where those integers are not necessarily distinct (i.e., duplicates are allowed in a binary heap). Here you must find a non-empty sequence of n integers to be inserted into h, such that each of those n insertions takes Θ(1) time. each of any subsequent extractmin() operations takes Θ(1) time. To describe your solution to this problem, place 15 suitable integers in the tree nodes shown below for a heap of size n = 15, as if they had already been inserted: (b) (5 points) The heaps we have studied are nearly complete because they fill from left-to-right, top-to-bottom, so that any unoccupied nodes occur toward the bottom-right of the tree. In those heaps, a node i s parent, left child, and right child were at index i, 2i, and 2i+1, respectively. 2 Imagine an analogous Hebrew 1 heap that fills right-to-left, top-to-bottom. A Hebrew heap of integers is shown below: We still want the array to fill as before, with the occupied portions of a Hebrew heap stored contiguously in the array, and with the heap s minimum element at index 1. If i is the index of a node in a Hebrew heap, fill in the blanks below to show the index of the specified related node: i s parent: i s left child: i s right child: 1 The Hebrew language is written and read right-to-left.

9 CSE 247 (Exam I) 9 Due by End of session 6. (10 points) Letters for you to use to fill in the blanks: a Θ(1) b Θ(logn) c Θ(n) d Θ(nlogn) e Θ(n 2 ) f linear search g binary search For each statement below, fill in the blank with a letter chosen from the above list. Correct responses may or may not be unique. The time to locate an integer in an array of size n using is O(logn). The time to locate an integer in an array of size 1,000,000,000,000,000 using binary search is. Thetimetoappnitemstoanemptylistwithnotailreferenceis. The timetoappnitems toanempty list withatailreference is. The time to perform n extractmin() operations in a heap initially of size n is. Thetimetoperform100extractMin()operationsinaheapinitiallyofsize n 100 is. Thetimetoinsertanintegeratthebeginningofalinkedlistofsizenis. The time to insert an integer at the beginning of an array list of size n is. If an exam contains n questions and it takes Θ(1) time to grade each question, then grading the exam takes time. If you are given a pile of n exams sorted alphabetically by last name, then the time taken to find your exam takes time using.

10 CSE 247 (Exam I) 10 Due by End of session