Final: CSS 342 SAMPLE. Data Structures, Algorithms, and Discrete Mathematics I

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1 Final: CSS 342 SAMPLE Data Structures, Algorithms, and Discrete Mathematics I This is a closed-book, closed-notes exam, except for a single sheet of letter-sized, double-sided hand-written notes. Total of 100 points, 120 minutes allowed time. Final covers all material in the course, but emphasis on topic covered after the midterm Midterm : Chapter 1-9 in Carrano + Appendix A, E, K + Interlude lectures + assignments Final : Chapter in Carrano, "An Active Introduction to Discrete Mathematics and Algorithms" by Cusack Chapter 7, "Cracking the Coding Interview" by G L McDowell Chapter 6 on Big O notation + lectures + assignments Algorithm Efficiency & Analysis 1. Review sample problems from lectures, "Cracking the Coding Interview" Chapter-6 and "An Active Introduction to Discrete Mathematics and Algorithms" page For the following functions, what is the upper bound O(?), and lower bound Ω(?), and tight bound Θ(?). Explain your analysis n 2 + n n n 2 + 5n 2.3. n 2 - n n 3-10n 2 + 5n n - 2n 3. What is the complexity of the following function? void testcomplexity1( int n) { int count = 0 ; for ( int i = 0 ; i < n; ++i) { for ( int j = i; j < n*n; ++j) { for ( int k = 0 ; k < 100 ; ++k) { 4. What is the complexity of the following function? 1

2 void testcomplexity2( int n) { int count = 0 ; for ( int i = 0 ; i < n; ++i) { for ( int k = 0 ; k < 100 ; ++k) { for ( int j = 1 ; j < n * n; ++j) { for ( int k = 0 ; k < 100 ; ++k) { 5. What is the complexity of the following function? void testcomplexity3(int n) { int count = 0 ; for (int i = 0 ; i < n; ++i) { for (int k = 0 ; k < 100 ; ++k) { for (int j = 1 ; j < n; j = j / 2 ) { Sorting Algorithms 1. What is the best case, and worst case complexity for the following algorithms: selection sort, bubble sort, insertion sort, merge sort, quick sort? 2. Which of the above algorithms would work best when the list is already sorted? Why? 3. Comparing two objects has a cost and swapping two objects in an array has a cost. Let's say comparing items costs 10 units and swapping costs 1 unit. How does this impact the performance of selection sort and bubble sort? 4. What is the complexity of bubblemergesort given below? You can assume that bubblesort and merge are implemented similar to their default implementation. void bubblemergesort(int thearray[], int first, int last) { if (first < last) { 2

3 int mid = first + (last - first) / 2 ; // midpoint bubblesort(thearray, first, mid); bubblesort(thearray, mid + 1, last); merge(thearray, first, mid, last); 5. What is the complexity of kthsmallest given below? You can assume that partition has a complexity of O(n). // This function returns k'th smallest element in arr[l..r] using QuickSort based method. ASSUMPTION: ALL ELEMENTS IN ARR[] ARE DISTINCT int kthsmallest(int arr[], int l, int r, int k) { // If k is smaller than number of elements in array if (k > 0 && k <= r - l + 1) { // Partition the array around last element and get // position of pivot element in sorted array int pos = partition(arr, l, r); // If position is same as k if (pos-l == k-1) return arr[pos]; if (pos-l > k-1) // If position is more, recur for left subarray return kthsmallest(arr, l, pos-1, k); // Else recur for right subarray return kthsmallest(arr, pos+1, r, k-pos+l-1); // If k is more than number of elements in array return INT_MAX; Queues & Priority Queues 1. Implement a priority queue for Square objects using an array as the underlying data structure. Square object only has a single variable called size. You should put actual Square objects, not pointers to Square objects, into your array. You can assume that the number of objects will be less than 100. class Square { 3

4 class Square { public : static int numberofsquares ; int size ; Square() : size { 0 { ++ numberofsquares ; ; explicit Square( int size) : size {size {++ numberofsquares ;; bool operator <( const Square &sq) { return size < sq. size ; ; int Square ::numberofsquares = 0 ; class PQueue { private : static const int MAX_SIZE = 100 ; int count { 0 ; public : PQueue() { Square thequeue [ MAX_SIZE ]; bool Add( Square sq) { if ( count == MAX_SIZE ) return false ; int start = 0 ; while (sq < thequeue [start] && start < count ) { ++start; if (start == count ) { thequeue [ count ] = sq; else { shiftelements(start); thequeue [start] = sq; ++ count ; return true ; void shiftelements( int start) { for ( int i = count ; i >= start; --i) { thequeue [i + 1 ] = thequeue [i]; 2. Implement a priority queue for Square objects using a doubly linked list as the underlying data structure. 4

5 Trees 1. Given a tree data type, search for an element in the tree class ATree { public : ATree() { bool Contains( int data) { return findnode(data, rootptr )!= nullptr ; private : struct Node { int data ; Node * left ; Node * right ; ; Node * rootptr { nullptr ; Node * findnode( int data, Node * startnode) { if (startnode == nullptr ) return nullptr ; else if (data == startnode-> data ) return startnode; else if (data < startnode-> data ) return findnode(data, startnode-> left ); else return findnode(data, startnode-> right ); ; 2. Find the smallest/largest element in a binary search tree? If the tree is balanced what is the complexity of this operation? If the tree is unbalanced, what is worst case complexity of this operation? 3. Add a new element to ATree structure given above 4. Write a class function for ATree, called CopyToTree, that returns a dynamically created array with all the ATree elements in it from smallest to largest. You will need to use helper functions and pass an array and a counter by reference to helper function. (Difficult) 5

6 Propositional Logic 1. Construct the truth table for the following propositions: 1.1. p AND (q OR (r IMPLIES p)) 1.2. p OR (q AND (r IFF p)) 1.3. (p IMPLIES q) IFF (q IMPLIES p) 2. Simplify the following propositions: 2.1. NOT (p AND q) 2.2. NOT (p IMPLIES q) 2.3. NOT (p IFF q) 2.4. (p AND q) OR (p IMPLIES q) OR (NOT (p AND q)) 3. Simplify the following C++ expressions 3.1. (x => 0) && (!(y < x)) && (! (x == y)) 3.2. (! ((x < 0) && (y > 0)) ((x > 0) && (y < 0)) ) 4. Given an array arr with length n, write the expression that will be true if either a) the array is an odd length: the first, last and middle element is the same b) the array is an even length: the first, the two middle elements and last element is same ( 3, 7, 10, 3, 3, 9, 8, 3 ) 6