Previous TableofContents Next  Copyright © CRC Press LLC  Algorithms and Data Structures in C++ by Alan Parker CRC Press, CRC Press LLC ISBN: 0849371716 Pub Date: 08/01/93  Previous  TableofContents Next 2.3.4TowerofHanoi The Tower of Hanoi problem is illustrated in Figure 2.1. The problem is to move n discs (in this case, three) from the first peg, A, to the third peg, C. The middle peg, B, may be used to store discs during the transfer. The discs have to be moved under the following condition: at no time may a disc on a peg have a wider disc above it on the same peg. As long as the condition is met all three pegs may be used to complete the transfer. For example the problem may be solved for the case of three by the following move sequence: where the ordered pair, (x, y), indicates to take a disk from peg x and place it on peg y. Figure 2.1 Tower of Hanoi Problem The problem admits a nice recursive solution. The problem is solved in terms of n by noting that to move n discs from A to C one can move n - 1 discs from A to B move the remaining disc from A to C and then move the n - 1 discs from B to C. This results in the relation for the number of steps, S (n), required for size n as with the boundary conditions Eq. 2.25 admits a solution of the form and matching the boundary conditions in Eq. 2.26 one obtains A growing field of interest is the visualization of algorithms. For instance, one might want to animate the solution to the Tower of Hanoi problem. Each disc move results in a new picture in the animation. If one is to incorporate the pictures into a document then a suitable language for its representation is PostScript. 1 This format is supported by almost all word processors and as a result is encountered frequently. A program to create the PostScript® description of the Tower of Hanoi is shown in Code List 2.6 The program creates an encapsulated postscript file shown in Code List 2.7. The word processor used to generate this book took the output of the program in Code List 2.7 and imported it to yield Figure 2.1! This program illustrates many features of C++.  1 PostScript®isatrademarkofAdobeSystemsInc.  The program utilizes only a small set of the PostScript® language. This primitive subset is described in Table 2.3. Table2.3PostScript®— PrimitiveSubsetCommand  Description xsetgray setthegrayleveltox.x=1iswhiteandx=0isblack.Thiswillaffectthe filloperation.  xyscale scaletheXdimensionbyxandscaletheYdimensionbyy. xsetlinewidth setthelinewidthtox. xymoveto startasubpathandmovetolocationxyonthepage. xyrlineto drawalinefromcurrentlocation(x 1 ,y 1 )to(x 1 +x,y 1 +y).Makethe endpointthecurrentlocation.Appendsthelinetothesubpath.  fill closethesubpathandfilltheareaenclosed. newpath createanewpathwithnocurrentpoint. showpage displaysthepagetotheoutputdevice. The program uses a number of classes in C++ which are derived from one another. This is one of the most powerful concepts in object-oriented programming. The class structure is illustrated in Figure 2.2. In the figure there exists a high-level base class called the graphic context. In a typical application a number of subclasses might be derived from it. In this case the graphics context specifies the line width, gray scale, and scale for its subsidiary objects. A derived class from the graphics context is the object class. This class contains information about the position of the object. This attribute is common to objects whether they are rectangles, circles, etc. A derived class from the object class is the rectangle class. For this class, specific information about the object is kept which identifies it with a rectangle, namely the width and the height. The draw routine overrides the virtual draw function for the object. The draw function in the object class is void even though for more complex examples it might have a number of operations. The RECTANGLE class inherits all the functions from the GRAPHICS_CONTEXT class and the OBJECT class. In the program, the rectangle class instantiates the discs, the base, and the pegs. Notice in Figure 2.1 that the base and pegs are drawn in a different gray scale than the discs. This is accomplished by the two calls in main(): •peg.set_gray(0.6) •base.set_gray(0.6) Any object of type RECTANGLE defaults to a set_gray of 0.8 as defined in the constructor function for the rectangle. Notice that peg is declared as a RECTANGLE and has access to the set_gray function of the GRAPHICS_CONTEXT. The valid operations on peg are: •peg.set_line_width(),fromtheGRAPHICS_CONTEXTclass •peg.set_scale(),fromtheGRAPHICS_CONTEXTclass •peg.set_gray(),fromtheGRAPHICS_CONTEXTclass •peg.location(),fromtheOBJECTclass •peg.set_location(),fromtheRECTANGLEclass •peg.set_width(),fromtheRECTANGLEclass •peg.set_height(),fromtheRECTANGLEclass •peg.draw(),fromtheRECTANGLEclass The virtual function draw in the OBJECT class is hidden from peg but it can be accessed in C++ using the scoping operator with the following call: •peg.object::draw(),usesdrawfromtheOBJECTclass Previous TableofContents Next  Copyright © CRC Press LLC  Algorithms and Data Structures in C++ by Alan Parker CRC Press, CRC Press LLC ISBN: 0849371716 Pub Date: 08/01/93  Previous  TableofContents Next Hence, in the program, all the functions are available to each instance of the rectangle created. This availability arises because the functions are declared as public in each class and each derived class is also declared public. Without the public declarations C++ will hide the functions of the base class from the derived class. Similarly, the data the functions access are declared as protected which makes the data visible to the functions of the derived classes. The first peg in the program is created with rectangle peg(80,0,40,180). The gray scale for this peg is changed from the default of 0.8 to 0.6 with peg.set_gray(0.6). The peg is drawn to the file with peg.draw(file). This draw operation results in the following lines placed in the file: •newpath •1setlinewidth •0.6setgray •800moveto •0180rlineto •400rlineto •0‐180rlineto •fill The PostScript® action taken by the operation is summarized in Figure 2.3. Note that the rectangle in the figure is not drawn to scale. The drawing of the base and the discs follows in an analogous fashion. Code List 2.6 Program to Display Tower of Hanoi Figure 2.2 Class Structure Figure 2.3 PostScript Rendering Code List 2.7 File Created by Program in Code List 2.6 2.3.5BooleanFunctionImplementation This section presents a recursive solution to providing an upper bound to the number of 2-input NAND gates required to implement a boolean function of n boolean variables. The recursion is obtained by noticing that a function, f(x 1 ,x 2 , ,x n ) of n variables can be written as for some functions g and h of n - 1 boolean variables. The implementation is illustrated in Figure 2.4. The number of NAND gates thus required as a function of n, C (n), can be written recursively as:  . TableofContents Next  Copyright © CRC Press LLC  Algorithms and Data Structures in C+ + by Alan Parker CRC Press, CRC Press LLC ISBN: 0849 371 716 Pub Date: 08/01/93  Previous  TableofContents. the graphics context specifies the line width, gray scale, and scale for its subsidiary objects. A derived class from the graphics context is the object class. This class contains information. the following call: •peg.object::draw(),usesdrawfromtheOBJECTclass Previous TableofContents Next  Copyright © CRC Press LLC  Algorithms and Data Structures in C+ + by Alan