University of Waterloo Department of Electrical and Computer Engineering ECE 250 Algorithms and Data Structures

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1 Uiversity of Waterloo Departmet of Electrical ad Computer Egieerig ECE 250 Algorithms ad Data Structures Midterm Examiatio ( pages) Istructor: Douglas Harder February 7, :30-9:00 Name (last, first) Studet ID Do problems -7. The last questio is a bous. The umber i brackets deotes the weight of the questio. If iformatio appears to be missig from a problem, make a reasoable assumptio, state it, ad proceed. If the space to aswer a questio is isufficiet, use the back of the previous page. You may use diagrams to supplemet (but ot replace) setece aswers. Closed book. No calculators. Questio Bous Total Mark /0 /2 /0 /6 /2 /0 /6 /3 /66 p of

2 Algorithm Aalysis. [0] Use the detailed model, that is, use τ +, τ fetch, etc., to determie the ru-time of the body of the followig method. Use the give table ad do ot total the ru times. You may use τ F, τ S, τ C, τ R, ad τ N, i place of τ fetch, τ store, τ call, τ retur, ad τ ew, respectively. You may wish to recall that the time it takes to fetch the istace variable array.legth is equal to the time it takes to fetch a local variable. public void average( it[] array ) { it sum = 0; 2 if ( array.legth == 0 ) 3 retur 0; 4 for ( it i = 0; i < array.legth; i ++ ) 5 sum += array[i]; 6 retur sum / array.legth; Lie Case (describe) Case 2 (describe) 2 3 4a 4b 4c 5 6 p 2 of

3 Asymptotic Aalysis 2a. [3] From the defiitio of big-o, show that = O(2 2 ). 2b. [2] Show that l() = Θ( log 2 () ), that is, l( ) ad log 2 ( ) are big-o of each other. f ( ) 2c. [2] Suppose lim =. For each of these statemets, circle true if is correct, ad = g ( ) false otherwise. f() = O( g() ) true false g() = O( f() ) true false 2d. [3] Prove that x = O( e x ) where is a fixed positive iteger. 2e. [2] Simplify the expressio O( ) + O( ) T( f ) + O( l( ) ) + O( ) where T( f ) is the time it takes to make a call to the method f. p 3 of

4 Foudatioal Data Structures 3. [5+5] Write the code for the methods preped, which iserts a object ito the start of the liked list; ad getcout, which couts the umber of elemets i the liked list ad returs that value. public class LikedList { protected Elemet head; protected Elemet tail; public fial class Elemet { Object datum; Elemet ext; public Elemet( Object datum, Elemet ext ) { this.datum = datum; this.ext = ext; public void preped( Object obj ) { public it getcout() { p 4 of

5 Abstract Data Types 4. [6] A cotaier is kow to have zero or more objects of the wrapper class Double. Write a method average which takes such a cotaier as a argumet, fids the average of all the elemets, ad returs that average. Your method should call the geteumeratio method of the cotaier ad use the retured eumeratio to access the elemets i the cotaier. If the cotaier is empty, your method should throw the CotaierEmptyExceptio exceptio. You eed ot check that the elemets i the cotaier are istaces of the class Double. public iterface Eumeratio { public boolea hasmoreelemets(); public Object extelemet(); public class Double { double value; public double doublevalue() { retur value; public SomeClass { public double average( Cotaier c ) { p 5 of

6 Stacks ad Queues 5a. [3+3] Suppose the class StackAsLikedList implemets a stack usig a siglyliked list which cotais a head referece ad a tail referece. For example: public class LikedList { protected Elemet head, tail; public class Elemet { Object datum; Elemet ext; The class LikedList has two methods which may be used to isert a elemet ito the liked list: apped, which iserts a ew elemet at the tail of the liked list; ad preped, which iserts a ew elemet at the head of the liked list. Therefore, there are two possible implemetatios of the push method of a stack: public void pushapped( Object o ) { list.apped( o ); public void pushpreped( Object o ) { list.preped( o ); If the method pushapped is chose to be used i the implemetatio of the push method of the class StackAsLikedList, what must the ru time, usig asymptotic otatio, of the pop method for a stack cotaiig elemets? Why (give oe setece)? If the method pushpreped is chose, what is the ru time, usig asymptotic otatio, of the pop method for a stack cotaiig elemets? Why (give oe setece)? p 6 of

7 5b. [6] For the class AbstractQueue, write the method appedqueue which takes as its argumet ay object which implemets the Queue iterface. This method should dequeue the elemets i the argumet queue ad equeue them i the curret queue util either the argumet queue is empty or this queue is full. Whe this method returs, all elemets i the argumet queue must either be i this queue (i the same order i which they appeared i the argumet queue) or must be i argumet queue (agai i the order i which they were placed ito the queue). public abstract class AbstractQueue exteds AbstractCotaier implemets Queue { public void appedqueue( Queue q ) { public iterface Queue exteds Cotaier { Object gethead(); void equeue( Object object ); Object dequeue(); public iterface Cotaier exteds Comparable { it getcout(); boolea isempty(); boolea isfull(); void purge(); void accept ( Visitor visitor ); Eumeratio geteumeratio(); p 7 of

8 Ordered ad Sorted Lists 6a. [3] Circle which elemets would be examied i the followig implemetatio of a sorted list of itegers whe determiig if the value 42 is stored i this array b. [2] Why ca you ot perform a biary search o a sorted list which is implemeted as a liked list? Use oe or two seteces. 6c. [3] A biary search rus i O( l( ) ) time where is the umber of elemets i the sorted list. You may fid this by solvig the recurrece relatio T( ) = T( ( )/2 ) +, ad T( ) =. Suppose istead you have a algorithm which, at each step, checks the midpoit of ot oe, but both halves of the sorted array. The recurrece relatio for such a problem is give by T( ) = 2 T( ( )/2 ) +, ad T() =. Assumig that = 2 k, what is the ru time of such a algorithm? Your aswer should be sufficietly simple to justify your ext aswer. 6d. [2] Justify coceptually why the solutio to the previous recurrece relatio makes sese. Use oe or two seteces. p 8 of

9 Project 2 Cosider the followig implemetatio of method SetCoefficiet i the class PolyomialAsArray. The array of polyomial coefficiets is stored i the istace variable coeffs. public void SetCoefficiet( it i, double c ) { if ( i < 0 ) throw ew RutimeExceptio(); 2 elif ( i < this.getdegree() ) coeffs[i] = c; 3a elif ( i == this.getdegree() ) { 3b coeffs[i] = c; 3c if ( c == 0 ) 3d resize_coeffs(); // shrik the array 4a else if ( c!= 0 ) { // i > this.getdegree() 4b double[] tmp = ew double[i + ]; 4c tmp[i] = c; 4d for ( it j = 0; j <= this.getdegree(); j++ ) 4e tmp[j] = this.coeffs[j]; 4f coeffs = tmp; What is the rutime of the followig two implemetatios of the assig method which sets the curret polyomial equal to the argumet polyomial p? You should refer to the code above, but you eed ot do more tha a cursory asymptotic aalysis. You may use to represet the degree of the polyomial p. public void assig( Polyomial p ) { coeffs = ew double[0]; 7a. [3] for ( it i = 0; i <= p.getdegree(); i++ ) this.setcoefficiet( i, p.getcoefficiet( i ) ); p 9 of

10 public void assig( Polyomial p ) { coeffs = ew double[0]; 7b. [3] for ( it i = p.getdegree(); i >= 0; i-- ) this.setcoefficiet( i, p.getcoefficiet( i ) ); Bous. [3] If is the maximum of the degree of this polyomial ad the degree of the polyomial p, the the average rutime for the followig method is O( ). Uder what coditios o this ad p will the rutime of this algorithm be O( 2 )? public Polyomial add( polyomial p ) { Polyomial q = ew PolyomialAsArray(); Polyomial mi, max; if ( p.getdegree() > this.getdegree() ) { max = p; mi = this; else { max = this; mi = p; for ( it i = max.getdegree(); i >= 0; i-- ) { q.setcoefficiet( i, max.getcoefficiet(i) ); for ( it i = mi.getdegree(); i >= 0; i-- ) { q.setcoefficiet( i, mi.getcoefficiet(i) + q.getcoefficiet(i) ); retur q; p 0 of

11 Additioal Iformatio d dx e x = e x d l( x) = dx x d dx x = x d dx x m ( 2 ) =! = 2 2 τ fetch, τ store, τ +, τ, τ *, τ /, τ <, τ call, τ retur, τ [ ], τ ew i= i= i= 0 i i ( + )(2 = ( i = = ) ( r ) r ) + i ir = 2 i= 0 ( r ) + r p of