Taking the Vector Class on a Random Walk

Listing 11.15 provides a short program that uses the revised Vector class. It simulates the famous Drunkard’s Walk problem. Actually, now that drunks are recognized as people with a serious health problem rather than as a source of amusement, it’s usually called the Random Walk problem. The idea is that you place someone at a lamppost. The person begins walking, but the direction of each step varies randomly from the direction of the preceding step. One way of phrasing the problem is this: How many steps does it take the random walker to travel, say, 50 feet away from the post? In terms of vectors, this amounts to adding a bunch of randomly oriented vectors until the sum exceeds 50 feet.

Listing 11.15 lets you select the target distance to be traveled and the length of the wanderer’s step. It maintains a running total that represents the position after each step (represented as a vector) and reports the number of steps needed to reach the target distance, along with the walker’s location (in both formats). As you’ll see, the walker’s progress is quite inefficient. A journey of 1,000 steps, each 2 feet long, may carry the walker only 50 feet from the starting point. The program divides the net distance traveled (50 feet, in this case) by the number of steps to provide a measure of the walker’s inefficiency. All the random direction changes make this average much smaller than the length of a single step. To select directions randomly, the program uses the standard library functions rand(), srand(), and time(), described in the following “Program Notes” section. Be sure to compile Listing 11.14 along with Listing 11.15.

Listing 11.15. randwalk.cpp

// randwalk.cpp -- using the Vector class
// compile with the vect.cpp file
#include <iostream>
#include <cstdlib>      // rand(), srand() prototypes
#include <ctime>        // time() prototype
#include "vect.h"
int main()
{
    using namespace std;
    using VECTOR::Vector;
    srand(time(0));     // seed random-number generator
    double direction;
    Vector step;
    Vector result(0.0, 0.0);
    unsigned long steps = 0;
    double target;
    double dstep;
    cout << "Enter target distance (q to quit): ";
    while (cin >> target)
    {
        cout << "Enter step length: ";
        if (!(cin >> dstep))
            break;

        while (result.magval() < target)
        {
            direction = rand() % 360;
            step.reset(dstep, direction, Vector::POL);
            result = result + step;
            steps++;
        }
        cout << "After " << steps << " steps, the subject "
            "has the following location: ";
        cout << result << endl;
        result.polar_mode();
        cout << " or " << result << endl;
        cout << "Average outward distance per step = "
            << result.magval()/steps << endl;
        steps = 0;
        result.reset(0.0, 0.0);
        cout << "Enter target distance (q to quit): ";
    }
    cout << "Bye! ";
    cin.clear();
    while (cin.get() != ' ')
        continue;
    return 0;
}

Because the program has a using declaration bringing Vector into scope, the program can use Vector::POL instead of VECTOR::Vector::POL.

Here is a sample run of the program in Listings 11.13, 11.14, and 11.15:

Enter target distance (q to quit): 50
Enter step length: 2
After 253 steps, the subject has the following location:
(x,y) = (46.1512, 20.4902)
 or
(m,a) = (50.495, 23.9402)
Average outward distance per step = 0.199587
Enter target distance (q to quit): 50
Enter step length: 2
After 951 steps, the subject has the following location:
(x,y) = (-21.9577, 45.3019)
 or
(m,a) = (50.3429, 115.8593)
Average outward distance per step = 0.0529362
Enter target distance (q to quit): 50
Enter step length: 1
After 1716 steps, the subject has the following location:
(x,y) = (40.0164, 31.1244)
 or
(m,a) = (50.6956, 37.8755)
Average outward distance per step = 0.0295429
Enter target distance (q to quit): q
Bye!

The random nature of the process produces considerable variation from trial to trial, even if the initial conditions are the same. On average, however, halving the step size quadruples the number of steps needed to cover a given distance. Probability theory suggests that, on average, the number of steps (N) of length s needed to reach a net distance of D is given by the following equation:

N = (D/s)2

This is just an average, but there will be considerable variations from trial to trial. For example, 1,000 trials of attempting to travel 50 feet in 2-foot steps yielded an average of 636 steps (close to the theoretical value of 625) to travel that far, but the range was from 91 to 3,951. Similarly, 1,000 trials of traveling 50 feet in 1-foot steps averaged 2,557 steps (close to the theoretical value of 2,500), with a range of 345 to 10,882. So if you find yourself walking randomly, be confident and take long steps. You still won’t have any control over the direction you wind up going, but at least you’ll get farther.

Program Notes

First, let’s note how painless it was to use the VECTOR namespace in Listing 11.15. The following using declaration places the name of the Vector class in scope:

using VECTOR::Vector;

Because all the Vector class methods have class scope, importing the class name also makes the Vector methods available, without the need for any further using declarations.

Next, let’s talk about random numbers. The standard ANSI C library, which also comes with C++, includes a rand() function that returns a random integer in the range from 0 to some implementation-dependent value. The random walk program uses the modulus operator to get an angle value in the range 0 to 359. The rand() function works by applying an algorithm to an initial seed value to get a random value. That value is used as the seed for the next function call, and so on. The numbers are really pseudorandom because 10 consecutive calls normally produce the same set of 10 random numbers. (The exact values depend on the implementation.) However, the srand() function lets you override the default seed value and initiate a different sequence of random numbers. This program uses the return value of time(0) to set the seed. The time(0) function returns the current calendar time, often implemented as the number of seconds since some specific date. (More generally, time() takes the address of a type time_t variable and puts the time into that variable and also returns it. Using 0 for the address argument obviates the need for an otherwise unneeded time_t variable.) Thus, the following statement sets a different seed each time you run the program, making the random output appear even more random:

srand(time(0));

The cstdlib header file (formerly stdlib.h) contains the prototypes for srand() and rand(), whereas ctime (formerly time.h) contains the time() prototype. (C++11 provides more extensive random number support with functions supported by the random header file.)

The program uses the result vector to keep track of the walker’s progress. On each cycle of the inner loop, the program sets the step vector to a new direction and adds it to the current result vector. When the magnitude of result exceeds the target distance, the loop terminates.

By setting the vector mode, the program displays the final position in rectangular terms and in polar terms.

Incidentally, the following statement has the effect of placing result in the RECT mode, regardless of the initial modes of result and step:

result = result + step;

Here’s why. First, the addition operator function creates and returns a new vector that holds the sum of the two arguments. The function creates that vector by using the default constructor, which creates vectors in the RECT mode. Thus, the vector being assigned to result is in the RECT mode. By default, assignment assigns each member variable individually, so RECT is assigned to result.mode. If you would prefer some other behavior, such as result retaining its original mode, you can override default assignment by defining an assignment operator for the class. Chapter 12 shows examples of this.

By the way, it’s a simple matter to save successive positions in a file. First, you include <fstream>, declare an ofstream object, and associate the object with a file:

#include <fstream>
...
ofstream fout;
fout.open("thewalk.txt");

Then, in the loop that calculates the result, you insert something like this:

fout << result << endl;

This invokes the friend function call operator<<(fout, result), causing the os reference parameter to refer to fout, thus sending output to the file. You could also use fout to write other information to the file, such as the summary information currently displayed by cout.

Automatic Conversions and Type Casts for Classes

The next topic on the class menu is type conversion. We’ll look into how C++ handles conversions to and from user-defined types. To set the stage, let’s first review how C++ handles conversions for its built-in types. When you make a statement that assigns a value of one standard type to a variable of another standard type, C++ automatically converts the value to the same type as the receiving variable, provided that the two types are compatible. For example, the following statements all generate numeric type conversions:

long count = 8;      // int value 8 converted to type long
double time = 11;    // int value 11 converted to type double
int side = 3.33;     // double value 3.33 converted to type int 3

These assignments work because C++ recognizes that the diverse numeric types all represent the same basic thing—a number—and because C++ incorporates built-in rules for making the conversions. Recall from Chapter 3, “Dealing with Data,” however, that you can lose some precision in these conversions. For example, assigning 3.33 to the int variable side results in side getting the value 3, losing the 0.33 part.

The C++ language does not automatically convert types that are not compatible. For example, the following statement fails because the left side is a pointer type, whereas the right side is a number:

int * p = 10;  // type clash

And even though a computer may represent an address internally with an integer, integers and pointers are conceptually quite different. For example, you wouldn’t square a pointer. However, when automatic conversions fail, you may use a type cast:

int * p = (int *) 10;  // ok, p and (int *) 10 both pointers

This sets a pointer to the address 10 by type casting 10 to type pointer-to-int (that is, type int *). Whether this assignment makes sense is another matter.

You may define a class sufficiently related to a basic type or to another class that it makes sense to convert from one form to another. In such a case, you can tell C++ how to make such conversions automatically or, perhaps, via a type cast. To see how that works, you can recast the pounds-to-stone program from Chapter 3 into class form. First, you need to design an appropriate type. Fundamentally, you’re representing one thing (a weight) two ways (pounds and stone). A class provides an excellent way to incorporate two representations of one concept into a single entity. Therefore, it makes sense to place both representations of weight into the same class and then provide class methods for expressing the weight in different forms. Listing 11.16 provides the class header.

Listing 11.16. stonewt.h

// stonewt.h -- definition for the Stonewt class
#ifndef STONEWT_H_
#define STONEWT_H_
class Stonewt
{
private:
    enum {Lbs_per_stn = 14};      // pounds per stone
    int stone;                    // whole stones
    double pds_left;              // fractional pounds
    double pounds;                // entire weight in pounds
public:
    Stonewt(double lbs);          // constructor for double pounds
    Stonewt(int stn, double lbs); // constructor for stone, lbs
    Stonewt();                    // default constructor
    ~Stonewt();
    void show_lbs() const;        // show weight in pounds format
    void show_stn() const;        // show weight in stone format
};
#endif

As mentioned in Chapter 10, enum provides a convenient way to define class-specific constants, provided that they are integers. Or you could use the following alternative:

static const int Lbs_per_stn = 14;

Note that the Stonewt class has three constructors. They allow you to initialize a Stonewt object to a floating-point number of pounds or to a combination of stone and pounds. Or you can create a Stonewt object without initializing it:

Stonewt blossem(132.5);   // weight = 132.5 pounds
Stonewt buttercup(10, 2); // weight = 10 stone, 2 pounds
Stonewt bubbles;          // weight = default value

The class doesn’t really need to declare a destructor because the automatic default constructor is fine for this case. On the other hand, providing an explicit declaration prepares you for the future, when you will have to define constructors.

Also the Stonewt class provides two display functions. One displays the weight in pounds, and the other displays the weight in stone and pounds. Listing 11.17 shows the class methods implementation. Note that each constructor assigns values to all three private members. Thus, creating a Stonewt object automatically sets both representations of weight.

Listing 11.17. stonewt.cpp

// stonewt.cpp -- Stonewt methods
#include <iostream>
using std::cout;
#include "stonewt.h"

// construct Stonewt object from double value
Stonewt::Stonewt(double lbs)
{
    stone = int (lbs) / Lbs_per_stn;    // integer division
    pds_left = int (lbs) % Lbs_per_stn + lbs - int(lbs);
    pounds = lbs;
}

// construct Stonewt object from stone, double values
Stonewt::Stonewt(int stn, double lbs)
{
    stone = stn;
    pds_left = lbs;
    pounds =  stn * Lbs_per_stn +lbs;
}

Stonewt::Stonewt()          // default constructor, wt = 0
{
    stone = pounds = pds_left = 0;
}

Stonewt::~Stonewt()         // destructor
{
}

// show weight in stones
void Stonewt::show_stn() const
{
    cout << stone << " stone, " << pds_left << " pounds ";
}

// show weight in pounds
void Stonewt::show_lbs() const
{
    cout << pounds << " pounds ";
}

Because a Stonewt object represents a single weight, it makes sense to provide ways to convert an integer or a floating-point value to a Stonewt object. And you have already done so! In C++, any constructor that takes a single argument acts as a blueprint for converting a value of that argument type to the class type. Thus the following constructor serves as instructions for converting a type double value to a type Stonewt value:

Stonewt(double lbs);  // template for double-to-Stonewt conversion

That is, you can write code like the following:

Stonewt myCat;        // create a Stonewt object
myCat = 19.6;         // use Stonewt(double) to convert 19.6 to Stonewt

The program uses the Stonewt(double) constructor to construct a temporary Stonewt object, using 19.6 as the initialization value. Then memberwise assignment copies the contents of the temporary object into myCat. This process is termed an implicit conversion because it happens automatically, without the need of an explicit type cast.

Only a constructor that can be used with just one argument works as a conversion function. The following constructor has two arguments, so it cannot be used to convert types:

Stonewt(int stn, double lbs);  // not a conversion function

However, it would act as a guide to int conversion if it provided a default value for the second parameter:

Stonewt(int stn, double lbs = 0);  // int-to-Stonewt conversion

Having a constructor work as an automatic type-conversion function seems like a nice feature. As programmers acquired more experience working with C++, however, they found that the automatic aspect isn’t always desirable because it can lead to unexpected conversions. So C++ added a new keyword, explicit, to turn off the automatic aspect. That is, you can declare the constructor this way:

explicit Stonewt(double lbs);   // no implicit conversions allowed

This turns off implicit conversions such as the preceding example but still allows explicit conversions—that is, conversions using explicit type casts:

Stonewt myCat;          // create a Stonewt object
myCat = 19.6;           // not valid if Stonewt(double) is declared as explicit
mycat = Stonewt(19.6);  // ok, an explicit conversion
mycat = (Stonewt) 19.6; // ok, old form for explicit typecast

When does the compiler use the Stonewt(double) function? If the keyword explicit is used in the declaration, Stonewt(double) is used only for an explicit type cast; otherwise, it is also used for the following implicit conversions:

• When you initialize a Stonewt object to a type double value

• When you assign a type double value to a Stonewt object

• When you pass a type double value to a function that expects a Stonewt argument

• When a function that’s declared to return a Stonewt value tries to return a double value

• When any of the preceding situations use a built-in type that can unambiguously be converted to type double

Let’s look at the last point in more detail. The argument-matching process provided by function prototyping lets the Stonewt(double) constructor act as conversions for other numerical types. That is, both of the following statements work by first converting int to double and then using the Stonewt(double) constructor:

Stonewt Jumbo(7000);   // uses Stonewt(double), converting int to double
Jumbo = 7300;          // uses Stonewt(double), converting int to double

However, this two-step conversion process works only if there is an unambiguous choice. That is, if the class also defined a Stonewt(long) constructor, the compiler would reject these statements, probably pointing out that an int can be converted to either a long or a double, so the call is ambiguous.

Listing 11.18 uses the class constructors to initialize some Stonewt objects and to handle type conversions. Be sure to compile Listing 11.17 along with Listing 11.18.

Listing 11.18. stone.cpp

// stone.cpp -- user-defined conversions
// compile with stonewt.cpp
#include <iostream>
using std::cout;
#include "stonewt.h"
void display(const Stonewt & st, int n);
int main()
{
    Stonewt incognito = 275; // uses constructor to initialize
    Stonewt wolfe(285.7);    // same as Stonewt wolfe = 285.7;
    Stonewt taft(21, 8);

    cout << "The celebrity weighed ";
    incognito.show_stn();
    cout << "The detective weighed ";
    wolfe.show_stn();
    cout << "The President weighed ";
    taft.show_lbs();
    incognito = 276.8;      // uses constructor for conversion
    taft = 325;             // same as taft = Stonewt(325);
    cout << "After dinner, the celebrity weighed ";
    incognito.show_stn();
    cout << "After dinner, the President weighed ";
    taft.show_lbs();
    display(taft, 2);
    cout << "The wrestler weighed even more. ";
    display(422, 2);
    cout << "No stone left unearned ";
    return 0;
}

void display(const Stonewt & st, int n)
{
    for (int i = 0; i < n; i++)
    {
        cout << "Wow! ";
        st.show_stn();
    }
}

Here is the output of the program in Listing 11.18:

The celebrity weighed 19 stone, 9 pounds
The detective weighed 20 stone, 5.7 pounds
The President weighed 302 pounds
After dinner, the celebrity weighed 19 stone, 10.8 pounds
After dinner, the President weighed 325 pounds
Wow! 23 stone, 3 pounds
Wow! 23 stone, 3 pounds
The wrestler weighed even more.
Wow! 30 stone, 2 pounds
Wow! 30 stone, 2 pounds
No stone left unearned