University of the Western Cape Department of Computer Science

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University of the Western Cape Department of Computer Science Algorithms and Complexity CSC212 Paper II Final Examination 13 November 2015 Time: 90 Minutes. Marks: 100.

1 University of the Western Cape Department of Computer Science Algorithms and Complexity CSC212 Paper II Final Examination 13 November 2015 Time: 90 Minutes. Marks: 100. UWC number Surname, first name Mark out of 100 Solution No calculators or cellphones are allowed. Please answer all the questions directly on this question paper. 1. Calculate the closest integer in decimal that is represented by S = 8 r=2 2r. 8 S = 2 r (1) r=2 = (2) = (3) = (4) = (5) = 508 (6) [2] Answer only Give the closest rational fraction of the form p/q, represented by 8 r=2 2 r. S = 8 2 r (1) r=2 = (2) = (3) = (4) = 127/256 (5) [2] Answer only 127/ What is the largest positive unsigned number that can be stored in eight bits? = = = 255. Answer only 255. [1] 4. What is the largest positive signed number that can be stored in eight bits? = = = 127. Answer only 127. [1]

2 5. How many different characters can be stored using eight bits? 2 8 = 256. Answer only 256. [2] 6. A 32-bit IEEE 754 floating-point number is divided into three regions, namely its sign, and its exponential and its fractional parts. How many bits are occupied by the exponential part? Give n for the greatest value of the fractional part in the form of 2 n 1 if it were to be stored as an integer. Eight, n = 7. A bit is needed for the sign. [1] 7. Given the array A[ ] = [23,17,5,9,11,7,19,29,31,37], give the value of the pivot, A[p], and the value, p, returned by the code snippet from quicksort: p = partition(a, 0, 9); quicksort(a, 0, p-1); immediately after partition has returned from its second call. p = 5,A[5] = 19. [2][ 8. You are given the array A[ ] = [17,5,9,11,7,19,23,37]. Give the exact number of comparisons needed by linear search to discover that the number 23 is in the list. 7. [2] 9. You are given the array A[ ] = [5,7,9,11,17,19,23,37]. How many times must the above array be halved by binary search to discover that the number 10 is not in the list. 4. [2] 10. You are given the array A[ ] = [5,7,9,11,17,19,23,37,41]. How many times must the above array be halved by binary search to discover that the number 37 is in the list. 2. [2] 11. You are given a list of numbers in this order A[ ] = [23,37,5,7,9,11,17,19]. How many comparisons are needed to turn this list into a maximizing priority heap using the bottom-up algorithm? 7. [2] 12. Suppose that a rather inefficient algorithm is used to find the 2nd largest element in an array of numbers A 0,A 1,...,A N. It does this by repeatedly removing the smallest element from the array, until only two elements are left over. The smallest is the required number. Let N = 6. Give the exact number of comparisons required. [2] An efficient method of getting the 2nd largest element, is by first building a maximizing heap from the bottomupandthenselectingthesecondlargestelement. If the heap has already been built, howmany comparisons are required to find the 2nd largest element in an array of numbers A 0,A 1,...,A N?[2] If a heap is built from the bottom up in an array of numbers A 0,A 1,...,A N with the largest element at A 0. How many comparisons are required to find the 3rd largest element? Explain. [2] 3. The second largest element A s, where s = 1 or s = 2, is identified by comparing the two children of A 0 comparison one. First get the biggest child of A s comparison two, and then compare this with the sibling A 3 s of A s comparison three. 15. How long does it take to build an N element heap top-down? Write your answer as O(...). [2] O(N logn) building a heap bottom-up takes O(N) time.

3 16. Supposing you are required to find the median, which is the middle element of a sorted list with an odd number of elements, otherwise it is the average of the two elements nearest the middle. Given a sorted int array A[ ] that has N +1 elements stored in positions A[0], A[1],..., A[N]. In Java, give an if statement that stores the value of the median in the variable that has been declared using int median; Give Java code for the if statement. [4] if (N % 2 == 0) // then N + 1 is odd return A[N/2]; else return (A[N/2] + A[N/2 + 1])/2; A mark is deducted for each Java mistake, leaving out a semicolon, writing N 2, etc Given the following code int moves = 0; static void hanoi(int n, int from, int other, int to) { moves++; if (n == 1) System.out.println("move " + from + " to " + to); else { hanoi (n-1, from, to, other); hanoi (1, from, other, to); hanoi (n-1, other, from, to); } } 18. What is the value of moves after hanoi(9, 1,2,3); has been executed? [3] n moves total moves = 766 N 2 N 2 N Calculating the powerset of a set with N elements takes Θ(2 n ) time. Briefly describe another problem that has the same time complexity. [2] Simply looping from 1 to 2 N. An N bit odometer, starting at , , , ,..., , , runs to The moves of the towers of Hanoi. Breaking an N letter password can take 26 N attempts. Travelling salesman. Find some that are not mentioned here.

4 4 20. Suppose a divide-and-conquer algorithm has an input of order N, and the time it takes to execute is T(N). If the method splits the input into two parts and requires a time of N to combine these two results then T(N) = 2T(N/2) + cn. Since there is nothing to do when N = 1, assume that T(1) = 0. Derive a value for T(N). Suppose that f(n) is some function of N, give an expression in the form of O(f(N)) that gives the time complexity of this divide-and-conquer algorithm. [4] Extracted from notes. Solving the recursive equation for merge sort is quite easy. T(N) = 2T(N/2)+cN. T(N) = 2T(N/2)+cN = 2[2T(N/4)+cN/2]+cN = 2 2 T(N/2 2 )+cn +cn = 2 2 [2T(N/2 3 )+cn/2 2 ]+cn +cn = 2 3 T(N/2 3 )+cn +cn +cn = 2 3 T(N/2 3 )+3cN. = 2 k T(N/2 k )+kcn Since there is nothing to do when the input is of size 1, it follows that T(1) = 0, so we set N/2 k = 1 giving N = 2 k. This happens when k = log 2 N and T(N) = T(1)+kcN = kcn = cn log 2 N = O(N logn). 21. For the sake of this question, suppose that you have a computer that is capable of executing instructions of time complexity Θ(1) in one nano second. Without using a calculator estimate the time in years that a program with time complexity Θ(4 N ) will take to complete when N = 32. Show all

5 your calculations below and draw a box around your answer. No box no marks. [4] Extracted from notes. Some problems are theoretically impossible to program. Other problems take interminably long. An O(2 n ) problem takes too long to run to completion using a super fast nanosecond computer that runs O(1) problems in one nanosecond when n is as small as ns = (2 10 ) 10 ns = ns ns = ns = seconds = hours = days = years = years years ns = years The O(4 n ) problem on a nanosecond per unit computer: 4 32 ns = (2 2 ) 32 ns = 2 64 ns = 2 4 (2 10 ) 6 = ns ns = ns = seconds = hours = days = years years = years = 500years. By approximating with 1000 instead of 1024 reduces the answer. Replacing the dividend = by also reduces the estimate. Your calculator which is banned from the examination should give = years. 22. Carefully describe an example of a program that has a time complexity of Θ(n n ). [4] An N digit base 10 odometer can count 10 N clicks, so a base N odometer with N digits can count N N clicks. 23. Give a concise iterative algorithm in pseudocode for depth-first search in a directed graph. [6] 24. Give an algorithm in pseudocode for doing topological sorting. [8] 25. Assuming that the graph weights are all positive, give Dijkstra s single-source-shortest-path algorithm for finding the length of shortest path from node v in a graph G = (V,E) to every other node u in the graph. [10]

6 26. Give Kruskal s algorithm for constructing a minimal spanning tree in a weighted graph. [10] What is the halting problem? [3] 28. What is meant by partially computable? Give an example of something that is not even partially computable. [3] 29. What is meant by NP-complete? [3] 30. What are the practical implications of knowing that a problem is NP-complete. [3] 31. Give five examples of NP-complete problems. [5] Total [100]

CS 171: Introduction to Computer Science II. Quicksort

CS 171: Introduction to Computer Science II Quicksort Roadmap MergeSort Recursive Algorithm (top-down) Practical Improvements Non-recursive algorithm (bottom-up) Analysis QuickSort Algorithm Analysis Practical