Section 5.5. Numeric Processing Examples

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5.5. Numeric Processing Examples

In this section we consider several numeric programming examples. They have been carefully chosen to illustrate different issues and concepts associated with processing numeric data.

5.5.1. Example: Rounding to Two Decimal Places

As an example of how to use Math class methods, we begin with the problem of rounding numbers. When dealing with applications that involve monetary valuesdollars and centsit is often necessary to round a calculated result to two decimal places. For example, suppose a program computes the value of a certificate of deposit (CD) to be 75.19999. Before we output this result, we would want to round it to two decimal places75.20. The following algorithm can be used to accomplish this:

1. Multiply the number by 100, giving 7519.9999. 2. Add 0.5 to the number giving 7520.4999. 3. Drop the fractional part giving 7520 4. Divide the result by 100, giving 75.20 

Algorithm design

Step 3 of this algorithm can be done using the Math.floor(R) method, which rounds its real argument, R, to the largest integer not less than R (from Table 5.11). If the number to be rounded is stored in the double variable R, then the following expression will round R to two decimal places:

R = Math.floor(R * 100.0 + 0.5) / 100.0; 

Alternatively, we could use the Math.round() method (Table 5.11). This method rounds a floating-point value to the nearest integer. For example, Math.round(65.3333) rounds to 65 and Math.round(65.6666) rounds to 66. The following expression uses it to round to two decimal places:

R = Math.round(100.0 * R) / 100.0; 

Note that it is important here to divide by 100.0 and not by 100. Otherwise, the division will give an integer result and we will lose the two decimal places.

Debugging Tip: Division

Using the correct type of literal in division operations is necessary to ensure that you get the correct type of result.

5.5.2. Example: Converting Fahrenheit to Celsius

To illustrate some of the issues that arise in using numeric data, we will now design a program that performs temperature conversions from Fahrenheit to Celsius, and vice versa.

Problem Decomposition

This problem requires two classes, a Temperature class and a TemperatureUI class. The Temperature class will perform the temperature conversions, and TemperatureUI will serve as the user interface (Fig. 5.3).

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Figure 5.3. Interacting objects: The user interacts with the user interface (TemperatureUI), which interacts with the Temperature object.

What objects do we need?

Class Design: Temperature

The purpose of the Temperature class is to perform the temperature conversions. To convert a Celsius temperature to Fahrenheit, or vice versa, it is not necessary to store the temperature value. Rather, a conversion method could take the Celsius (or Fahrenheit) temperature as a parameter, perform the conversion, and return the result. Therefore, the Temperature class does not need any instance variables. In this respect the Temperature class resembles the Math class. Unlike OneRowNim, which stores the game's statethe number of sticks remaining and whose turn it isthe Math and Temperature classes are stateless.

What data do we need?

Thus, following the design of the Math class, the Temperature class will have two public static methods: one to convert from Fahrenheit to Celsius, and one to convert from Celsius to Fahrenheit. Recall that static methods are associated with the class rather than with its instances. Therefore, we need not instantiate a Temperature object to use these methods. Instead, we can invoke the methods through the class itself.

What methods do we need?

The methods will use the standard conversion formulas: and . Each of these methods should have a single parameter to store the temperature value that is being converted.

Because we want to be able to handle temperatures such as 98.6, we will use real-number data for the methods' parameters. Generally speaking, the double type is more widely used than float because Java represents real literals such as 98.6 as doubles. Because doubles are more widely used in Java, using double wherever a floating-point value is needed will cut down on the number of implicit data conversions that a program would have to perform. Therefore, each of our conversion methods should take a double parameter and return a double result. These considerations lead to the design shown in Figure 5.4.

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Figure 5.4. The Temperature class. Note that static elements are underlined in UML.

Java Programming Tip: Numeric Types

Java uses the int type for integer literals and double for real-number literals. When possible, using int and double for numeric variables and parameters reduces the number of implicit conversions a program would have to perform.

Implementation: Temperature

The implementation of the Temperature class is shown in Figure 5.5. Note that because celsToFahr() uses the double value temp in its calculation, it uses floating-point literals (9.0, 5.0, and 32.0) in its conversion expression. This helps to reduce the reliance on Java's built-in promotion rules, which can lead to subtle errors. For example, suppose we had written what looks like an equivalent calculation using integer literals:

return(9 / 5 * temp + 32); // Error: equals (temp + 32) 

Figure 5.5. The Temperature class.

public class Temperature {     public Temperature() {}     public static double fahrToCels(double temp)     {         return (5.0 * (temp - 32.0) / 9.0);     }     public static double celsToFahr(double temp)     {         return (9.0 * temp / 5.0 + 32.0);     } } // Temperature class 

Because 9 divided by 5 gives the integer result 1, this expression is always equivalent to temp + 32, which is not the correct conversion formula. This kind of subtle semantic error can be avoided if you avoid mixing types wherever possible.

Semantic error

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Java Programming Tip: Don't Mix Types

You can reduce the incidence of semantic errors caused by implicit type conversions if, whenever possible, you explicitly change all the literals in an expression to the same type.

Testing and Debugging

The next question to be addressed concerns the way to test this program. As always, the program should be tested in a stepwise fashion. As each method is coded, you should test it both in isolation and in combination with the other methods, if possible.

Testing strategy

In addition, you should develop appropriate test data. It is not enough to just plug in any values. The values you use should test for certain potential problems. For this program, the following tests are appropriate:

• Test converting 0 degrees C to 32 degrees F.

• Test converting 100 degrees C to 212 degrees F.

• Test converting 212 degrees F to 100 degrees C.

• Test converting 32 degrees F to 0 degrees C.

Designing test data

The first two tests use the celsToFahr() method to test the freezing-point and boiling-point temperatures, two boundary values for this problem. A boundary value is a value at the beginning or end of the range of values that a variable or calculation is meant to represent. The second pair of tests performs similar checks with the fahrToCels() method. One advantage of using these values is that we know what results the methods should return.

Effective Design: Test Data

Developing appropriate test data is an important part of program design. One type of test data should check the boundaries of the calculations you are making.

Debugging Tip: Test, Test, Test!

The fact that your program runs correctly on some data is no guarantee of its correctness. The more testing, and the more careful the testing, the better.

The TemperatureUI Class

The purpose of the TemperatureUI class is to serve as a user interfacethat is, as an interface between the user and a Temperature object. It will accept a Fahrenheit or Celsius temperature from the user, pass it to one of the public methods of the Temperature object for conversion, and display the result that is returned.

As we discussed in Chapter 4, the user interface can take various forms, ranging from a command-line interface to a graphical interface. Figure 5.6 shows a design for the user interface based on the command-line interface developed in Chapter 4. The TemperatureUI uses a KeyboardReader to handle interaction with the user and uses static methods in the Temperature class to perform the temperature conversions.

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Figure 5.6. A command-line user interface.

Self-Study Exercises

Exercise 5.14

Following the design in Figure 5.6, implement the TemperatureUI class and use it to test the methods in Temperature class. The run() method should use an input-process-output algorithm: Prompt the user for input, perform the necessary processing, and output the result. Since Temperature's conversion methods are class methods, you do not need to instantiate a Temperature object in this project. You can invoke the conversion methods directly through the Temperature class: double fahr = Temperature.celsToFahr(98.6);

Exercise 5.15

Following the design for the GUI developed in Chapter 4, implement a GUI to use for testing the Temperature class. The GUI should have the layout shown in Figure 5.7. Figure 5.7. Layout design of a GUI that performs temperature conversions.

5.5.3. Example: Using Class Constants

As we noted in Chapter 0, in addition to instance variables, which are associated with instances (objects) of a class, Java also allows class variables, which are associated with the class itself. One of the most common uses of such variables is to define named constants to replace literal values. A named constant is a variable that cannot be changed once it has been given an initial value. In this section, we use our running example, OneRowNim, to illustrate using class constants.

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Recall that methods and variables associated with a class must be declared with the static modifier. If a variable is declared static, there will be exactly one copy of that variable created no matter how many times its class is instantiated. To turn a variable into a constant, it must be declared with the final modifier. Thus, the following are examples of class constantsconstant values associated with the class rather than with its instances:

public static final int PLAYER_ONE = 1; public static final int PLAYER_TWO = 2; public static final int MAX_PICKUP = 3; public static final int MAX_STICKS = 7; 

The final modifier indicates that the value of a variable cannot be changed. When final is used in a variable declaration, the variable must be assigned an initial value. After a final variable is properly declared, it is a syntax error to attempt to try to change its value. For example, given the preceding declarations, the following assignment statement would cause a compiler error:

PLAYER_ONE = 5; // Syntax error; PLAYER_ONE is a constant 

Note how we use uppercase letters and underscore characters (_) in the names of constants. This is a convention that professional Java programmers follow, and its purpose is to make it easy to distinguish the constants from the variables in a program. This makes the program easier to read and understand.

Java Programming Tip: Readability

To make your programs more readable, use uppercase font for constant identifiers.

Another way that named constants improve the readability of a program is by replacing the reliance on literal values. For example, for the OneRowNim class, compare the following two if conditions:

if (num < 1 || num > 3 || num > nSticks) ... if (num < 1 || num > MAX_PICKUP || num > nSticks) ... 

Clearly, the second condition is easier to read and understand. In the first condition, we have no good idea what the literal value 3 represents. In the second, we know that MAX_PICKUP represents the most sticks a player can pick up.

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Thus, to make OneRowNim more readable, we should replace all occurrences of the literal value 3 with the constant MAX_PICKUP. The same principle would apply to some of the other literal values in the program. Thus, instead of using 1 and 2 to represent the two players, we could use PLAYER_ONE and PLAYER_TWO to make methods such as the following easier to read and understand:

public int getPlayer() {   if (onePlaysNext)          return PLAYER_ONE;     else return PLAYER_TWO; } // getPlayer() 

Readability

Java Programming Tip: Readability

To make your programs more readable use named constants instead of literal values.

Another advantage of named constants (over literals) is that their use makes the program easier to modify and maintain. For example, suppose that we decide to change OneRowNim so that the maximum number of sticks that can be picked up is four instead of three. If we used literal values, we would have to change all occurrences of 4 that were used to represent the maximum pick-up. If we use a named constant, we need only change its declaration to:

public static final int MAX_PICKUP = 4; 

Maintainability

Effective Design: Maintainability

Constants should be used instead of literal values in a program. This will make the program easier to modify and maintain.

All of the examples we have presented so far show why named constants (but not necessarily class constants) are useful. Not all constants are class constants. That is, not all constants are declared static. However, the idea of associating constants with a class makes good sense. Creating just a single copy of the constant saves memory resources, and in addition constants such as MAX_STICKS and PLAYER_ONE make more conceptual sense when associated with the class itself rather than with any particular OneRowNim instance.

Class constants are used extensively in the Java class library. For example, as we saw in Chapter 2, Java's various built-in colors are represented as constants of the java.awt.Color classColor.blue and Color.red. Similarly, java.awt.Label uses int constants to specify how a label's text should be aligned: Label.CENTER.

Another advantage of class constants is that they can be used before instances of the class exist. For example, a class constant (as opposed to an instance constant) may be used during object instantiation:

OneRowNim game = new OneRowNim(OneRowNim.MAX_STICKS); 

Note how we use the name of the class to refer to the class constant. Of course, MAX_STICKS has to be a public variable in order to be accessible outside the class. MAX_STICKS has to be a class constant if it is to be used as a constructor argument, because at this point in the program there are no instances of OneRowNim. A new version of OneRowNim that uses class constants is shown in Figure 5.8.

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Figure 5.8. This version of OneRowNim uses named constants.

public class OneRowNim {   public static final int PLAYER_ONE = 1;     public static final int PLAYER_TWO = 2;     public static final int MAX_PICKUP = 3;     public static final int MAX_STICKS = 11;     public static final boolean GAME_OVER = false;      private int nSticks = MAX_STICKS;      private boolean onePlaysNext = true;      public OneRowNim()      {      } // OneRowNim() constructor1      public OneRowNim(int sticks)      {   nSticks = sticks;      } // OneRowNim() constructor2      public OneRowNim(int sticks, int starter)      {   nSticks = sticks;          onePlaysNext = (starter == PLAYER_ONE);      } // OneRowNim() constructor3      public boolean takeSticks(int num)      {   if (num < 1 || num > MAX_PICKUP || num > nSticks)              return false;                          // Error          else                                       // Valid move          {   nSticks = nSticks - num;              onePlaysNext = !onePlaysNext;              return true;          } // else      } // takeSticks()      public int getSticks()      {   return nSticks;      } // getSticks()      public int getPlayer()      {   if (onePlaysNext)              return PLAYER_ONE;          else return PLAYER_TWO;       } // getPlayer()       public boolean gameOver()       {   return (nSticks <= 0);       } // gameOver()       public int getWinner()       {   if (nSticks < 1)               return getPlayer();           else return 0;                            // Game is not over       } // getWinner()       public String report()       {   return ("Number of sticks left: " + getSticks()            + "\nNext turn by player " + getPlayer() + "\n");       } // report() }  // OneRowNim class 

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It is important to note that Java also allows class constants to be referenced through an instance of the class. Thus, once we have instantiated game, we can refer to MAX_STICKS with either OneRowNim.MAX_STICKS or game.MAX_STICKS.

Self-Study Exercise

Exercise 5.16

Implement a command-line interface class named KBTestOneRowNim that uses our new version of OneRowNim. Make use of MAX_STICKS and MAX_PICKUP in the user interface.

5.5.4. Object-Oriented Design: Information Hiding

The fact that the two new versions of OneRowNim developed in this chapter are backward compatible with the earlier version is due in large part to the way we have divided up its public and private elements. Because the new versions still present the same public interface, programs that use the OneRowNim class, such as the OneRowNimApp from Chapter 4 (Fig. 4.24), can continue to use the class without changing a single line of their code. To confirm this, see the Self-Study Exercise at the end of this section.

Preserving the public interface

Although we have made significant changes to the underlying representation of OneRowNim, the implementation detailsits data and algorithmsare hidden from other objects. As long as OneRowNim's public interface remains compatible with the old version, changes to its private elements won't cause any inconvenience to objects dependent on the old version. This ability to change the underlying implementation without affecting the outward functionality of a class is one of the great benefits of the information-hiding principle.

Information hiding

Effective Design: Information Hiding

In designing a class, other objects should be given access only to the information they need and nothing more.

The lesson to be learned here is that the public parts of a class should be restricted to whatever must be accessible to other objects. Everything else should be private. Things work better, in Java programming and in the real world, when objects are designed with the principle of information hiding in mind.

Self-Study Exercise

Exercise 5.17

To confirm that our new version of OneRowNim still works correctly with the user interfaces we developed in Chapter 4, compile and run it with OneRowNimApp (Fig. 4.24).

5.5.5. Example: A Winning Algorithm for One-Row Nim

Now that we have access to numeric data types and operators, we will develop an algorithm that can win the One-Row Nim game. Recall that in Chapter 4 we left things such that when the computer moves, it always takes one stick. Let's replace that strategy with a more sophisticated approach.

If you have played One-Row Nim, you have probably noticed that in a game with 21 sticks, you can always win the game if you leave your opponent with 1, 5, 9, 13, 17, or 21 sticks. This is obvious for the case of one stick. For the case where you leave your opponent five sticks, you can make a move that leaves the other player with one stick no matter what the other player does. For example, if your opponent takes one stick, you can take three; if your opponent takes two, you can take two; and if your opponent takes three, you can take one. In any case, you can win the game by making the right move if you have left your opponent with five sticks. The same arguments apply for the other values: 9, 13, 17, and 21.

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What relationship is common to the numbers in this set? Note that you always get 1 if you take the remainder after dividing each of these numbers by 4:

 1 % 4  == 1  5 % 4  == 1  9 % 4  == 1 13 % 4  == 1 17 % 4  == 1 21 % 4  == 1 

Thus, we can base our winning strategy on the goal of leaving the opponent with a number of sticks, N, such that N % 4 equals 1.

To determine how many sticks to take in order to leave the opponent with N, we need to use a little algebra. Let's suppose that sticksLeft represents the number of sticks left before our turn. The first thing we have to acknowledge is that if sticksLeft %4 == 1, then we have been left with 1, 5, 9, 13, and so on, sticks, so we cannot force a win. In that case, it doesn't matter how many sticks we pick up. Our opponent should win the game.

Now let's suppose that sticksLeft % 4 != 1, and let M be the number of sticks to pick up in order to leave our opponent with N, such that N % 4 == 1. Then we have the following two equations:

sticksLeft - M == N N % 4 == 1 

We can combine these into a single equation, which can be simplified as follows:

(sticksLeft - M) % 4 == 1 

If sticksLeft - M leaves a remainder of 1 when divided by 4, that means that sticksLeft - M is equal to some integer quotient, Q times 4 plus 1:

(sticksLeft - M) == (Q * 4) + 1 

By adding M to both sides and subtracting 1 from both sides of this equation, we get:

(sticksLeft - 1) == (Q * 4) + M 

The equation is saying that (sticksLeft - 1) % 4 == M. That is, when you divide sticksLeft-1 by 4, you will get a remainder of M, which is the number of sticks you should pick up. Thus, to decide how many sticks to take, we want to compute:

M == (sticksLeft -1) % 4 

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To verify this, let's look at some examples:

sticksLeft     (sticksLeft -1) % 4     sticksLeft  Before                                  After ----------------------------------------------------   9            (9-1) % 4 == 0         Illegal Move   8            (8-1) % 4 == 3         5   7            (7-1) % 4 == 2         5   6            (6-1) % 4 == 1         5   5            (5-1) % 4 == 0         Illegal Move 

The examples in this table show that when we use (sticksLeft-1) % 4 to calculate our move, we always leave our opponent with a losing situation. Note that when sticksLeft equals 9 or 5, we can't apply this strategy because it would lead to an illegal move.

Let us now convert this algorithm into Java code. In addition to incorporating our winning strategy, this move() method makes use of two important Math class methods:

public int move() { int sticksLeft = nim.getSticks();                      // Get number of sticks   if (sticksLeft % (nim.MAX_PICKUP + 1) != 1)            // If winnable     return (sticksLeft - 1) % (nim.MAX_PICKUP +1);   else {                                                 // Else pick random     int maxPickup = Math.min(nim.MAX_PICKUP, sticksLeft);     return 1 + (int)(Math.random() * maxPickup);   } } 

The move() method will return an int representing the best move possible. It begins by getting the number of sticks left from the OneRowNim object, which is referred to as nim in this case. It then checks whether it can win by computing (sticksLeft-1) % 4. Rather than use the literal value 4, however, we use the named constant MAX_PICKUP, which is accessible through the nim object. This is an especially good use for the class constant because it makes our algorithm completely generalthat is, our winning strategy will continue to work even if the game is changed so that the maximum pickup is five or six. The then clause computes and returns (sticksLeft-1) % (nim.MAX_PICKUP+1), but here again it uses the class constant.

The else clause would be used when it is not possible to make a winning move. In this case we want to choose a random number of sticks between one and some maximum number. The maximum number depends on how many sticks are left. If there are more than three sticks left, then the most we can pick up is three, so we want a random number between 1 and 3. However, if there are two sticks left, then the most we can pick up is two and we want a random number between 1 and 2. Note how we use the Math.min() method to decide the maximum number of sticks that can be picked up:

int maxPickup = Math.min(nim.MAX_PICKUP, sticksLeft); 

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The min() method returns the minimum value between its two arguments.

Finally, note how we use the Math.random() method to calculate a random number between 1 and the maximum:

1 + (int)(Math.random() * maxPickup); 

The random() method returns a real number between 0 and 0.999999that is, a real number between 0 and 1 but not including 1:

0 <= Math.random() < 1.0 

If we multiply Math.random() times 2, the result would be a value between 0 and 1.9999999. Similarly, if we multiplied it by 3, the result would be a value between 0 and 2.9999999. In order to use the random value, we have to convert it into an integer. We do this by using the (int) cast operator:

(int)(Math.random() * maxPickup); 

Recall that when a double is cast into an int, Java just throws away the fractional part. Therefore, this expression will give us a value between 0 and maxPickup-1. If maxPickup is 3, this will give a value between 0 and 2, whereas we want a random value between 1 and 3. To achieve this desired value, we merely add 1 to the result. Thus, using the expression

1 + (int)(Math.random() * maxPickup) 

gives us a random number between 1 and maxPickup, where maxPickup is either 1, 2, or 3, depending on the situation of the game at that point.

Self-Study Exercises

Exercise 5.18

Implement a class named NimPlayer that incorporates the move() method designed in this section. The class should implement the design shown in Figure 5.9. That is, in addition to the move() method, it should have an instance variable, nim, which will serve as a reference to the OneRowNim game. Its constructor method should take a OneRowNim parameter, allowing the NimPlayer to be given a reference when it is instantiated. Figure 5.9. The NimPlayer class.

Exercise 5.19

Modify OneRowNim's command-line interface to play One-Row Nim between the user and the computer, where the NimPlayer implemented in the preceding exercise represents the computer.