RYERSON UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE CPS 616 MIDTERM WINTER 2017

Transcription

1 RYERSON UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE CPS 616 MIDTERM WINTER 2017 NAME: STUDENT ID: INSTRUCTIONS This exam is 1 hours and 45 minutes long. This exam is out of 50 and is worth 15% of the course mark. This is a closed book exam. However, one double-sided letter-sized crib sheet is allowed. This exam is single-sided and has 6 pages including this front page. The first part of this exam is multiple choice. Please enter your answers to this part on the bubble sheet provided. Students writing at the Test Center should simply circle their answer in this booklet instead. The second part of this exam is short answer. Please answer all questions of this part directly on this exam.

2 CPS 616 W2017 MIDTERM 2 PART 1 20 MARKS - MULTIPLE CHOICE Instructions Please enter your answers on the bubble sheet with your name unless you are writing this exam at the Test Centre, in which case you should just circle your answer for each question in this booklet. Use pencil only for bubble sheets.. YOU CAN GUESS. MARKS WILL NOT BE DEDUCTED FOR FALSE ANSWERS. Questions 1. (2 marks) Is this graph: h g f e A. Directed and cyclical B. Directed and acyclical C. Undirected and cyclical D. Undirected and acyclical 2. (2 marks) Which one of these sequences of vertices is not a topological sort?: h g f e A. a b h g c d f e B. e f g h C. a g c e b d f h D. None of the above (i.e. they are all topological sorts) 3. (2 marks) Which one of these sequences of vertices was generated from a BFS (Breadth First Search) traversal of the graph on the left? Assume ties are broken by the alphabetical order of the vertices. h g f e A. f c d g e a b h B. a g b c f d e h C. b g a h c e f d D. None of the above (i.e. none was generated from a BFS traversal) 4. (2 marks) Which one of these sequences of vertices was not generated from a DFS (Depth First Search) traversal of the graph on the left? Assume ties are broken by the alphabetical order of the vertices. h g f e A. h b g c f d e a B. a g b h c e f d C. b g a c e f d h D. None of the above (i.e. they were all generated from a DFS traversal)

3 CPS 616 W2017 MIDTERM 3 5. (3 marks) Which is the most desirable complexity for an algorithm (i.e. which of these complexities provides the most hope that the algorithm will be fast)? A. O(n) B. θ(n) C. Ω(log n) D. o(n) 6. (3 marks) What is the exact cost of this algorithm as a function of n: myloops (n) for (i=0; i<=n; i++) for (j=n; j>i, j--) // the basic operation is here A. n(n+1)/2 B. n(n-1)/2 C. (n+1) 2 /2 D. (n+1)(n-1)/2 7. (3 marks) Which of the three claims (A, B, C) about the two code fragments underneath is false: // Fragment 1 result1 = 0 for i=1 to n do for j=1 to n do for k=1 to n do if i<j result1 = result1++; // Fragment 2 result2 = 0 for i=1 to n do for j=1 to n do if i<j for k=1 to n do result2 = result2++; A. The value of result1 at the end of the execution of fragment1 is the same as the value of result2 at the end of the execution of fragment 2. B. Fragments 1 and 2 have the same exact execution cost C. Fragments 1 and 2 have the same asymptotic execution cost 8. (3 marks) Which of these sequences is not a Gray code: A. 111,101, 100, 000, 001, 011, 010, 110 B. abc, bac, bca, cba, cab, acb C. 00,01,02,12,11,10,20,21,22 D. None of the above: they are all Gray codes

4 CPS 616 W2017 MIDTERM 4 PART 2 SHORT ANSWERS 30 MARKS - PLEASE WRITE YOUR ANSWERS DIRECTLY IN THIS EXAM 9. (30 marks) Here is a decrease by one algorithm to solve the knapsack problem // Description of items is stored in arrays of n elements. Integer weights[n] // weights[i] = weight of item i Integer values[n] // values[i] = vaue of item i // Set Knapsack (Integer capacity, Integer n) // capacity is the knapsack capacity // n = number of items to fit in the knapsack, n1 // This function returns a set containing the most valuable combination of items // which can fit in the knapsack Set Knapsack (Integer capacity, Integer n) { If n=1 If weights[1]<= capacity return {1} Else return {} // If item n does not fit, leave it out Integer weightn = weights[n] If weightn>capacity return Knapsack(capacity,n-1) // Best value when item n is included in knapsack Set included = {n} Knapsack(capacity-weightn, n-1) // Best value when item n is not included in knapsack Set excluded = Knapsack(capacity,n-1) If Worth(excluded) >= Worth(included) Else } return excluded return included // Integer Worth(SetofItems) // SetofItems is a set of items // This function returns the worth of a set of items Integer Worth(SetofItems) { Integer result = 0 For item in SetofItems do Result += values[item] Return result } For the analysis that follows, you will be assuming that the basic operation is an array reference weights[i] or values[i], i.e. you will be asked to count how many times an item is touched. a) (2 marks) In the code above, identify clearly the locations of the basic operation (for example you can circle them, or highlight them, or underline them, etc.) b) (1 mark) What is the size of this algorithm? (i.e. what variables will the cost of this algorithm be a function of?)

5 CPS 616 W2017 MIDTERM 5 c) (2 marks) What is the exact cost of the function Worth? i.e. how many times will its basic operation be executed when it is called with the parameter SetofItems? d) (4 marks) What is the best case cost of the Knapsack algorithm and in which situation does it happen? e) (3 marks) In which situation does the worst case cost occur? (i.e. in which situation does this algorithm do the most work, maximizing both the work of Knapsack and Worth?) f) (5 marks) Define recursively the worst case cost Kn of the Knapsack function for n items. Remember that you need to provide both the base case and the recurrence relation. Also do not forget to include the cost of the function Worth in your cost. Justify your answer (i.e. explain what each component of the formula represents)

6 CPS 616 W2017 MIDTERM 6 g) (8 marks) Make a few modifications to this algorithm to integrate the functionality of the Worth function directly into the Knapsack function in an efficient way that minimizes the number of references to the values array. You do not need to worry about precise syntax as long as your pseudocode resembles actual code in a reasonable way. You can use pairs in your pseudocode, i.e. data structures of the form (a,b). You would refer to the first element a as (a,b)[1] and the second element b as (a,b)[2] Set Knapsack (Integer capacity, Integer n) { If n=1 If weights[1]<= capacity return {1} Else return {} Integer weightn = weights[n] If weightn>capacity return Knapsack(capacity,n-1) Set included = {n} Knapsack(capacity-weightn, n-1) Set excluded = Knapsack(capacity,n-1) If Worth(excluded) >= Worth(included) return excluded Else return included } h) (5 marks) Define recursively the worst case cost In of your improved Knapsack function for n items. Remember that you need to provide both the base case and the recurrence relation Justify your answer (i.e. explain what each component of the formula represents)