CMSC 341 Data Structure Asymptotic Analysis Review

CMSC 341 Data Structure Asymptotic Analysis Review

<p>CMSC 341 Data Structure <br>Asymptotic Analysis Review </p><p>These questions will help test your understanding of the asymptotic analysis material discussed in class and in the text. These questions are only a study guide. Questions found here may be on your exam, although perhaps in a different format. Questions NOT found here may also be on your exam. </p><p>1. What&nbsp;is the purpose of asymptotic analysis? </p><p>2. Define “Big-Oh” using a formal, mathematical definition. </p><p>3. Let&nbsp;<em>T</em><sub style="top: 0.13em;">1</sub>(<em>x</em>) = <em>O</em>(<em>f</em>(<em>x</em>)) and <em>T</em><sub style="top: 0.13em;">2</sub>(<em>x</em>) = <em>O</em>(<em>g</em>(<em>x</em>)). Prove <em>T</em><sub style="top: 0.13em;">1</sub>(<em>x</em>) + <em>T</em><sub style="top: 0.13em;">2</sub>(<em>x</em>) = <em>O </em>(max(<em>f</em>(<em>x</em>), <em>g</em>(<em>x</em>))). 4. Let&nbsp;<em>T</em>(<em>x</em>) = <em>O</em>(<em>cf</em>(<em>x</em>)), where <em>c </em>is some positive constant. Prove <em>T</em>(<em>x</em>) = <em>O</em>(<em>f</em>(<em>x</em>)). 5. Let&nbsp;<em>T</em><sub style="top: 0.13em;">1</sub>(<em>x</em>) = <em>O</em>(<em>f</em>(<em>x</em>)) and <em>T</em><sub style="top: 0.13em;">2</sub>(<em>x</em>) = <em>O</em>(<em>g</em>(<em>x</em>)). Prove <em>T</em><sub style="top: 0.13em;">1</sub>(<em>x</em>) * <em>T</em><sub style="top: 0.13em;">2</sub>(<em>x</em>) = <em>O</em>(<em>f</em>(<em>x</em>) * <em>g</em>(<em>x</em>)) </p><p>6. Prove&nbsp;2<sup style="top: -0.46em;">n+1 </sup>= <em>O</em>(2<sup style="top: -0.46em;">n</sup>). </p><p>7. Prove&nbsp;that if <em>T</em>(<em>n</em>) is a polynomial of degree <em>x</em>, then <em>T</em>(<em>n</em>) = <em>O</em>(<em>n</em><sup style="top: -0.46em;"><em>x</em></sup>). </p><p>8. Number&nbsp;these functions in ascending (slowest growing to fastest growing) Big-Oh order: </p><p></p><ul style="display: flex;"><li style="flex:1"><strong>Number </strong></li><li style="flex:1"><strong>Big-Oh </strong></li></ul><p></p><p><em>O</em>(<em>n</em><sup style="top: -0.46em;">3</sup>) <br><em>O</em>(<em>n</em><sup style="top: -0.46em;">2 </sup>lg <em>n</em>) <br><em>O</em>(1) <br><em>O</em>(lg<sup style="top: -0.46em;">0.1 </sup><em>n</em>) <em>O</em>(<em>n</em><sup style="top: -0.46em;">1.01 </sup><em>O</em>(<em>n</em><sup style="top: -0.46em;">2.01 </sup><em>O</em>(2<sup style="top: -0.46em;"><em>n</em></sup>) </p><p>))</p><p><em>O</em>(lg <em>n</em>) <em>O</em>(<em>n</em>) <br><em>O</em>(<em>n </em>lg <em>n</em>) <em>O </em>(<em>n </em>lg<sup style="top: -0.46em;">5 </sup><em>n</em>) </p><p>1<br>9. Determine,&nbsp;for the typical algorithms that you use to perform calculations by hand, the running time to: </p><p>a. Add&nbsp;two N-digit numbers b. Multiply&nbsp;two N-digit numbers </p><p>10. What is the asymptotic performance of each of the following? <br>Select among: </p><p></p><ul style="display: flex;"><li style="flex:1">a<em>. O</em>(<em>n</em>) </li><li style="flex:1">b. <em>O</em>(<em>n</em><sup style="top: -0.46em;">2</sup>) </li><li style="flex:1">c. <em>O</em>(<em>n </em>lg <em>n</em>) d.&nbsp;<em>O</em>(<em>n</em><sup style="top: -0.46em;">3</sup>) </li><li style="flex:1">e. <em>O</em>(lg <em>n</em>) </li></ul><p></p><ul style="display: flex;"><li style="flex:1">f. <em>O</em>(1) </li><li style="flex:1">g. <em>O</em>(<em>n</em>!) </li></ul><p></p><p>h. None of these <br>(a) ___________ Squaring each element of an NxN matrix (b) ___________ Finding the smallest value in a sorted array of N integers (c) ___________ Finding a value in a sorted array using binary search (d) ___________ Pushing N elements onto a stack, then popping them and printing them (e) ___________ Finding the largest 3 values in an unsorted array <br>11. What is the asymptotic performance of the following Java code fragment? <br>Justify your answer. </p><p>for (int i = 0; i &lt; N; i++) {for (int j = 10; j &gt;= 0; j--) {int count = 1; while (count &lt; N) count *= 2; <br>}<br>}</p><p>2</p><p>CMSC 341 Data Structures <br>List Review Questions </p><p>Please refer to the textbook for List, ListNode, and ListIterator class definitions. These definitions are the same as those found in the class notes. You may assume that all member functions of these classes have been written and work properly when answering the questions below. These questions will help test your understanding of the list material discussed in class and in the text. These questions are only a study guide. Questions found here may be on your exam, although perhaps in a different format. Questions NOT found here may also be on your exam. </p><p>1. Write&nbsp;a new member function of the List class named <strong>ReversePrint </strong>that uses an iterator to display the elements of the List in reverse order. The data elements should be enclosed </p><p>in angle brackets (“&lt; &gt;”) and separated by commas. Do not construct a copy of the list </p><p>that is in reverse order, just use iterators. The prototype for <strong>ReversePrint( ) </strong>is shown below </p><p>void ReversePrint ( ); </p><p>2. Write&nbsp;a new function named <strong>Splice( ) </strong>whose prototype is shown below. <strong>Note that </strong></p><p><strong>Splice( )is not a member function of the List class. </strong>This function “splices” L2 into L1 at </p><p>the specified position (<strong>pos</strong>is a ListIterator over L1). If <strong>pos</strong>is past end, <strong>Splice( ) </strong>does </p><p>nothing. For example, suppose L1 = {1, 2, 3, 4, 5} and L2 = {10, 20, 30} and <strong>pos</strong>is constructed as list1.iterator( ) and then advanced twice (pos.next()) so it is positioned </p><p>before the “3”. Then the function call <strong>Splice( L1, L2, pos )</strong>; causes L1 to </p><p>become { 1, 2, 10, 20, 30,3, 4, 5} and L2 is unchanged. What is the asymptotic performance of <strong>Splice( )</strong>? Bear in mind that there are two lists which may be of different lengths. </p><p>public static&lt;T&gt; void Splice( List&lt;T&gt; L1, const List&lt;T&gt; L2, ListIterator&lt;T&gt; pos); </p><p>3. Complete&nbsp;the following table of Big-Oh, worst-case asymptotic time performance for the given operations and implementations of the List ADT. Give your answers in terms of n, the number of elements in the list. </p><p><strong>Operation </strong></p><p>insert (index, x ) ind( x ) </p><p></p><ul style="display: flex;"><li style="flex:1"><strong>Double Linked List </strong></li><li style="flex:1"><strong>ArrayList </strong></li></ul><p></p><p>remove( x ) makeEmpty add(x ) </p><p>1<br>4. Suppose&nbsp;you are provided with a set of N random numbers which are to be inserted into a sorted List (smallest to largest). What would be the worst-case asymptotic time performance for building the entire list? </p><p>5. One&nbsp;advantage of using a double-linked list is the availability of bi-directional list </p><p>iterators – iterators that move “forward” and move “backward”. Suppose that in order to </p><p>save memory, you (or your boss) decide to use a single linked list. Can a single linked list still support bi-directional iterators? If so, briefly describe how the iterator would move forward and backward. Be sure to include the Big-Oh performance of these operations. If bi-directional iterators are not possible with a single linked list, explain why not. </p><p>6. Describe&nbsp;the advantages (if any) and disadvantages (if any) of each of the List ADT implementation – singly linked list, doubly linked list, and arraylist. </p><p>7.Write the code for a new LinkedList method that removed the node AFTER the one specified by the parameter.&nbsp;The signature of this new method is provided below. </p><p>private AnyType <strong>removeAfter</strong>( Node&lt;AnyType&gt; p ) </p><p>2</p><p>CMSC 341 Data Structures Stack and Queue Review </p><p>These are some review questions on stacks. The class definitions for stack and queue are provided at the end of the questions. These questions will help test your understanding of the stack and queue material discussed in class and in the text. These questions are only a study guide. Questions found here may be on your exam, although perhaps in a different format. Questions NOT found here may also be on your exam. </p><p><strong>Stacks </strong></p><p>1. Using&nbsp;only the operations of the stack, write a static function that determines if a </p><p>string is a palindrome (i.e. reads the same backward and forward; e.g. “level”). The </p><p>prototype for this function is given below. </p><p>public static bool isPalindrome(String theString); </p><p>2. What&nbsp;is the output of the following code? </p><p>int [] values = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19 }; Stack&lt;Integer&gt; s = new Stack&lt;Integer&gt;( ); </p><p>for (int 1 = 0; i &lt; values.length; i++) s.push( values[ i ] ); </p><p>int n = 25; for (int i = 0; i &lt; 4; i++) {n += s.pop( ); <br>}for (int i = 0; i &lt; 2; i++) {n -= s.pop( ); <br>}System.out.println( n ); </p><p>3. Discuss the advantages and disadvantages if the text’s array implementation and the </p><p>lecture notes layered implementation of the stack ADT. At a minimum, consider the </p><p>asymptotic time performance of the <strong>isEmpty( ), pop( )</strong>and <strong>push( ) </strong></p><p>operations. <br>1<br>4. Using&nbsp;only the operations of the stack given in the class, write a Java function that returns a copy of the user specified stack. The prototype for the function is given below. </p><p>public static&lt;AnyType&gt; Stack&lt;AnyType&gt; CopyStack(Stack&lt;AnyType&gt; otherStack) </p><p><strong>Queues </strong></p><p>1. Using&nbsp;the operations of the stack and queue, write a static function that determines if </p><p>a string is a palindrome (i.e. reads the same backward and forward; e.g. “level”). The </p><p>prototype for this function is given below. </p><p>public static bool isPalindrome(String theString ); </p><p>2. Suppose&nbsp;that Q is an initially empty array-based queue of size 5. Show the values of the data members front and back after each statement has been executed. Indicate any errors that might occur. </p><p>Queue&lt;Character&gt; Q( 5 );&nbsp;front = _____&nbsp;back = _____ </p><p>Q.enqueue ( ‘A’ ); Q.enqueue ( ‘B’ ); Q.enqueue ( ‘C’ ); </p><p>char c = Q.dequeue( ); </p><p>Q.enqueue ( ‘A’ ); </p><p>front = _____&nbsp;back = _____ front = _____&nbsp;back = _____ front = _____&nbsp;back = _____ front = _____&nbsp;back = _____ front = _____&nbsp;back = _____ </p><p>3. Discuss&nbsp;the advantages and disadvantages of the link list and array-based implementations of a queue. </p><p>4. Describe three “real life” applications of a queue. </p><p>5. Explain&nbsp;how to implement a queue using two stacks. <br>2</p><p><strong>Definition of the Stack Class </strong></p><p>This is the definition of the array based stack from the text: </p><p>public class Stack&lt;AnyType&gt; {public Stack ( int capacity ); public boolean isEmpty( ); public boolean isFull( ); public void makeEmpty( ); public AnyType pop( ); public AnyType top( ); public void push( AnyType element ); private ArrayList&lt; AnyType &gt; theArray; private int topOfStack; <br>}</p><p><strong>Definition of the Queue Class </strong></p><p>This is the definition of the array based queue from the text: </p><p>public class Queue&lt;AnyType&gt; {public Queue( int capacity ); public boolean isEmpty( ); public boolean isFull( ); public AnyType getFront( ); public void makeEmpty( ); public AnyType dequeue( ); public void enqueue( AnyType element ); private ArrayList&lt; AnyType &gt; the Array private int currentSize, front, back; private void increment( int x ); <br>}</p><p>3</p><p>CMSC 341 Data Structures <br>General Tree Review </p><p>These questions will help test your understanding of the general tree material discussed in class and in the text. These questions are only a study guide. Questions found here may be on your exam, although perhaps in a different format. Questions NOT found here may also be on your exam. </p><p><strong>General Trees </strong></p><p>1. Define&nbsp;<em>tree. </em></p><p>2. Define&nbsp;<em>k-ary tree. </em></p><p>3. For&nbsp;any tree, T, define the following a. path&nbsp;in T b. length&nbsp;of a path in T c. height&nbsp;of a node in T d. depth&nbsp;of a node in T e. height&nbsp;of T f. depth&nbsp;of T g. external&nbsp;node h. internal&nbsp;node i. leaf </p><p>4. Given&nbsp;the drawing of an arbitrary tree, draw the first-child, next-sibling representation of the tree. </p><p>5. Given&nbsp;the first-child, next-sibling representation of a tree, draw the tree. 6. Prove&nbsp;that there are <em>n </em>– 1 edges in any tree with <em>n </em>nodes. 7. What&nbsp;is the worst-case Big-Oh performance for the <strong>insert, find </strong>and <strong>remove </strong>operations in a general tree? Why is this so? </p><p>8. Write a recursive member function of the “static K-ary” tree class that counts the number </p><p>of nodes in the tree. <br>1</p><p><strong>Binary Trees </strong></p><p>1. Define&nbsp;<em>binary tree</em>, <em>full binary tree, complete binary tree </em>and <em>perfect binary tree </em></p><p>2. Prove&nbsp;that a perfect binary tree of height <em>h </em>has 2<sup style="top: -0.46em;"><em>h </em></sup>leaf nodes. 3. Prove&nbsp;that a perfect binary tree of height <em>h </em>has 2<sup style="top: -0.46em;"><em>h</em>+1 </sup>– 1 nodes. 4. Prove&nbsp;that a full binary tree with <em>n </em>internal nodes has <em>n </em>+ 1 leaf nodes. 5. Prove&nbsp;that in any binary tree with <em>n </em>nodes there are <em>n </em>+1 “null pointers”. 6. Suppose&nbsp;that you have two traversals from the same binary tree. Draw the tree. pre-order: A D F G H K L P Q R W Z in-order: G&nbsp;F H K D L A W R Q P Z </p><p>7. Write&nbsp;a recursive member function of the BinaryTree class that counts the number of nodes in the tree. </p><p>8. Write&nbsp;a recursive member function of the BinaryTree class that counts the number of leaves in the tree. </p><p>9. Given&nbsp;the following binary tree containing integers, list the output from a <em>pre-order </em></p><p><em>traversal</em>, an <em>in-order traversal</em>, a <em>post-order traversal</em>, and a <em>level-order traversal </em>of the </p><p>tree. </p><p>87 <br>114 <br>56 </p><p>93 <br>66 <br>42 </p><p>123 <br>39 <br>17 </p><p>29 </p><p>2</p><p>CMSC 341 Data Structures Binary Search Tree Review </p><p>These questions will help test your understanding of the binary search tree material presented in class and in the text. These questions are only a study guide. Questions found here may be found on your exam, although perhaps in a different format. Questions NOT found here may also be on your exam. The necessary class definitions will be provided with your exam. Please refer to the BST and BST node class definitions in the text. </p><p>1. Define&nbsp;<strong>binary search tree</strong>. </p><p>2. Write&nbsp;the syntactically correct Java code for the private BST member function find(T x)whose function prototype is found below and cannot be changed. This private function is initially called by the public find(T x)with a reference to the root. If x is in the tree, return element in the node; otherwise return NULL. </p><p>private T find(BinaryNode&lt;T&gt; root, T x) </p><p>3. Describe&nbsp;how deletion is performed in a BST. Be sure to mention all relevant cases. 4. The&nbsp;number of comparisons to build a BST of <em>n </em>elements is <em>O</em>(<em>n</em><sup style="top: -0.46em;">2</sup>) in the worst case and <br><em>O</em>(<em>n </em>lg <em>n</em>) in the best case. Describe the best and worst case and explain why this is so. </p><p>5. Insert&nbsp;the following Java reserved words into an initially empty BST in the order given: </p><p><em>break, operator, if, typedef, else, case, while, do, return, byte, for, true, double, void. </em></p><p>a. Show&nbsp;the output from the pre-order, post-order, in-order, and level-order traversals of the BST. b. Which&nbsp;word is the predecessor of <em>while</em>? c. Which&nbsp;word is the successor of <em>for</em>? d. Draw&nbsp;the new tree that results from each of the following operations. Each problem is separate and begins with the tree created above. i. Insert&nbsp;<em>int </em>and <em>float</em>, in that order. </p><p>ii. Delete&nbsp;<em>double</em>, <em>case</em>, and <em>typedef</em>, in that order. </p><p>6. Prove&nbsp;that if a node in a BST has two children, its successor has at most one child. 7. Suppose&nbsp;the result of a post-order traversal of a BST is 1, 3, 2, 8, 10, 5, 4. Draw the original BST. </p><p>1</p><p>CMSC 341 Data Structures <br>Splay Tree Review </p><p>These questions will help test your understanding of the splay tree material presented in </p><p>class. Note that the splay tree presented in class is a “bottom-up” splay tree. These questions are </p><p>only a study guide. Questions found here may be found on your exam, although perhaps in a different format. Questions NOT found here may also be on your exam. The splay tree rules <strong>may </strong>be provided with your exam. Check with your instructor. Please refer to the Splay tree class definitions in the text. </p><p>1. Define&nbsp;<strong>splay tree. </strong></p><p>2. Describe&nbsp;the splay operation. 3. Show&nbsp;the result of inserting the values <strong>2, 1, 4, 5, 9, 3, 6, 7 </strong>into an empty splay tree. Show the tree at the end of each insertion. Show each rotation. </p><p>4. Explain,&nbsp;in English, how to insert a new element into a splay tree. Be sure to discuss the case in which a duplicate element is inserted. </p><p>5. Explain,&nbsp;in English, how to find an element in a splay tree. Bu sure to discuss the case in which the element is not found. </p><p>6. Explain,&nbsp;in English, how to remove an element from a splay tree. Be sure to discuss the case in which the element to be removed is not in the tree. </p><p>7. What does the following statement mean? In particular, what does “<em>m</em>” represent? </p><p>“A splay tree has <em>O</em>(<em>m </em>lg <em>n</em>) amoritized performance over a sequence of insert, remove </p><p>and find operations.” </p><p>8. For&nbsp;splay tree operations, we use amortized analysis rather than asymptotic analysis. a. What&nbsp;is amortized analysis? b. How&nbsp;is amortized analysis different than asymptotic analysis? c. Why&nbsp;must amortized analysis be used for splay trees? </p><p>CMSC 341 Data Structures <br>Red-Black Tree Review </p><p>These questions will help test your understanding of the Red-Black tree material discussed in class and in the text. These questions are only a study guide. Questions found here may be on your exam, although perhaps in a different format. Questions NOT found here may also be on your exam. The rotation diagrams for red-black trees <strong>may </strong>be provided with your exam. Check with your instructor. </p><p>1. Define&nbsp;<strong>Red-Black tree</strong>. List all Red-Black tree properties 2. Define&nbsp;the <strong>black height </strong>of a node, <em>x</em>. 3. What is the “Big-Oh” performance (in terms of the number of nodes in the tree) for the operations <strong>find, insert </strong>and <strong>remove </strong>for a red-black tree in the best, worst and averages cases? </p><p>4. What&nbsp;property of red-black trees is most significant in explaining the “Big-Oh” </p><p>performance for the operations <strong>find, insert </strong>and <strong>remove</strong>? </p><p>5. Prove&nbsp;that in any red-black tree with root <em>x</em>, there are at least <em>n </em>= 2<sup style="top: -0.46em;"><em>bh</em>(<em>x</em>) </sup>– 1 internal nodes where <em>bh</em>(<em>x</em>) is the black-height of <em>x</em>. </p><p>6. Prove&nbsp;that in any red-black tree, at least half the nodes on any path from the root to a leaf must be black. </p><p>7. Prove&nbsp;that in any red-black tree, no path from any node, N, to a leaf is more than twice as long as any other path from N to any other leaf. </p><p>8. Prove&nbsp;that if a black node has just one child, that child must be red. 9. Show&nbsp;the tree that results from inserting the values <strong>2, 1, 4, 5, 9, 3, 6, 7 </strong>into an initially empty red-black tree. Show the tree after each insertion. </p><p>1<br>10. Given the following Red-Black Tree, show the tree that results after deleting the node with value <strong>68 </strong>using bottom-up deletion. </p><p>87 <br>114 <br>56 </p><p>49 <br>166 </p><ul style="display: flex;"><li style="flex:1">40 </li><li style="flex:1">68 </li></ul><p></p><p>Represents a BLACK node Represents a RED node </p><p>2</p><p>CMSC 341 Data Structures <br>K-D Tree Review </p><p>These questions will help test your understanding of k-D trees. These questions are only a study guide. Some of these questions may appear on your exam, although perhaps in a different format. Question NOT found on this study guide may also appear on your exam. </p><p>1. What&nbsp;is the purpose of k-D trees? </p><p>2. Explain&nbsp;how a 2-D tree extends a binary search tree. 3. Insert&nbsp;the following pairs (in the order shown) into an initially empty 2-D tree. <br>(53, 14), (27, 28), (30, 11), (67, 51), (70, 3) <br>4. Insert&nbsp;the following triples (in the order shown) into an initially empty 3-D tree. <br>(16, 8, 12), (18, 7, 15), (10, 9, 3), (13, 8, 9), (20, 10, 6), (3, 10, 7), (12, 14, 6) </p><p>1</p><p>CMSC 341 Data Structures <br>Priority Queue Review </p><p>These questions will help test your understanding of the priority queue material discussed in class and in the text. These questions are only a study guide. Questions found here may be on your exam, although perhaps in a different format. Questions NOT found here may also be on your exam. </p><p>1. Define&nbsp;the following terms a. priority b. priority&nbsp;queue c. min&nbsp;binary heap d. partial&nbsp;ordering e. null&nbsp;path length in a binary tree f. leftist&nbsp;binary tree g. leftist&nbsp;heap </p><p>2. Insertion&nbsp;and deletion (of the minimum element) in a min binary heap are <em>O</em>(<em>lg n</em>) on average. Explain why this is so. </p><p>3. Finding&nbsp;the minimum element in a min binary heap is <em>O</em>(1) in the worst case. Explain why this is so. </p><p>4. Although&nbsp;a binary heap is conceptually a binary tree, it can be implemented as an array. <br>Explain why this is so. </p><p>5. In&nbsp;a min binary heap with N elements, what is the range of indices in which the largest element will be found? </p><p>6. Describe,&nbsp;in English, an algorithm to find the largest element in a min binary heap. What is the asymptotic worst-case performance of your algorithm? </p><p>7. Assume&nbsp;that the array representing a min binary heap contains the values 2, 8, 3, 10, 16, <br>7, 18, 13, 15. Show the contents of the array after inserting the value 4. </p><p>8. Assume&nbsp;that the array representing a min binary heap contains the values 2, 8, 3, 10, 16, <br>7, 18, 13, 15. Show the contents of the array after deleting the minimum element. </p><p>9. Show&nbsp;the array representing the min binary heap constructed using the initial values 18, <br>2, 13, 10, 15, 3, 7, 16, 8. </p><p>10. Prove that the largest element in a min binary heap is a leaf. </p><p>1<br>11. Prove that a complete binary tree is a leftist tree. 12. Prove for any leftist tree with N nodes, the number of nodes, R, on the rightmost path to a non-full node is given by <em>R </em>&lt; <em>lg</em>(<em>N </em>+ 1) </p><p>13. Given the drawing of a binary tree, determine if the tree is a leftist tree and if it is a leftist heap. Give reasons why or why not. </p><p>14. Given the drawings of the two leftist heaps, draw the leftist heap that results from merging them. </p><p>15. Describe how to perform the <strong>findMin, insert, </strong>and <strong>deleteMin </strong>operations on a leftist heap. 16. Describe a method for constructing a leftist heap from an initial set of <em>N </em>values. Your algorithm must run in <em>O</em>(<em>N</em>) time. </p>

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