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Subsections Pseudocode and Stepwise Refinement The 3N+1 Problem Coding, Testing, Debugging

### Section 3.2

## Algorithm Development

Programming is difficult (like many activities that are useful and worthwhile -- and like most of those activities, it can also be rewarding and a lot of fun). When you write a program, you have to tell the computer every small detail of what to do. And you have to get everything exactly right, since the computer will blindly follow your program exactly as written. How, then, do people write any but the most simple programs? It's not a big mystery, actually. It's a matter of learning to think in the right way.

A program is an expression of an idea. A programmer starts with a general idea of a task for the computer to perform. Presumably, the programmer has some idea of how to perform the task by hand, at least in general outline. The problem is to flesh out that outline into a complete, unambiguous, step-by-step procedure for carrying out the task. Such a procedure is called an "algorithm." (Technically, an algorithm is an unambiguous, step-by-step procedure that always terminates after a finite number of steps. We don't want to count procedures that might go on forever.) An algorithm is not the same as a program. A program is written in some particular programming language. An algorithm is more like the idea behind the program, but it's the idea of the steps the program will take to perform its task, not just the idea of the task itself. When describing an algorithm, the steps don't necessarily have to be specified in complete detail, as long as the steps are unambiguous and it's clear that carrying out the steps will accomplish the assigned task. An algorithm can be expressed in any language, including English. Of course, an algorithm can only be expressed as an actual program if all the details have been filled in.

So, where do algorithms come from? Usually, they have to be developed, often with a lot of thought and hard work. Skill at algorithm development is something that comes with practice, but there are techniques and guidelines that can help. I'll talk here about some techniques and guidelines that are relevant to "programming in the small," and I will return to the subject several times in later chapters.

### 3.2.1 Pseudocode and Stepwise Refinement

When programming in the small, you have a few basics to work with: variables, assignment statements, and input/output routines. You might also have some subroutines, objects, or other building blocks that have already been written by you or someone else. (Input/output routines fall into this class.) You can build sequences of these basic instructions, and you can also combine them into more complex control structures such as while loops and if statements.

Suppose you have a task in mind that you want the computer to perform. One way to proceed is to write a description of the task, and take that description as an outline of the algorithm you want to develop. Then you can refine and elaborate that description, gradually adding steps and detail, until you have a complete algorithm that can be translated directly into programming language. This method is called stepwise refinement, and it is a type of top-down design. As you proceed through the stages of stepwise refinement, you can write out descriptions of your algorithm in pseudocode -- informal instructions that imitate the structure of programming languages without the complete detail and perfect syntax of actual program code.

As an example, let's see how one might develop the program from the previous section, which computes the value of an investment over five years. The task that you want the program to perform is: "Compute and display the value of an investment for each of the next five years, where the initial investment and interest rate are to be specified by the user." You might then write -- or more likely just think -- that this can be expanded as:

    Get the user's input
    Compute the value of the investment after 1 year
    Display the value
    Compute the value after 2 years
    Display the value
    Compute the value after 3 years
    Display the value
    Compute the value after 4 years
    Display the value
    Compute the value after 5 years
    Display the value

This is correct, but rather repetitive. And seeing that repetition, you might notice an opportunity to use a loop. A loop would take less typing. More important, it would be more general: Essentially the same loop will work no matter how many years you want to process. So, you might rewrite the above sequence of steps as:

    Get the user's input
    while there are more years to process:
        Compute the value after the next year
        Display the value

Following this algorithm would certainly solve the problem, but for a computer we'll have to be more explicit about how to "Get the user's input," how to "Compute the value after the next year," and what it means to say "there are more years to process." We can expand the step, "Get the user's input" into

    Ask the user for the initial investment
    Read the user's response
    Ask the user for the interest rate
    Read the user's response

To fill in the details of the step "Compute the value after the next year," you have to know how to do the computation yourself. (Maybe you need to ask your boss or professor for clarification?) Let's say you know that the value is computed by adding some interest to the previous value. Then we can refine the while loop to:

    while there are more years to process:
        Compute the interest
        Add the interest to the value
        Display the value

As for testing whether there are more years to process, the only way that we can do that is by counting the years ourselves. This displays a very common pattern, and you should expect to use something similar in a lot of programs: We have to start with zero years, add one each time we process a year, and stop when we reach the desired number of years. This is sometimes called a counting loop. So the while loop becomes:

    years = 0
    while years < 5:
        years = years + 1
        Compute the interest
        Add the interest to the value
        Display the value

We still have to know how to compute the interest. Let's say that the interest is to be computed by multiplying the interest rate by the current value of the investment. Putting this together with the part of the algorithm that gets the user's inputs, we have the complete algorithm:

    Ask the user for the initial investment
    Read the user's response
    Ask the user for the interest rate
    Read the user's response
    years = 0
    while years < 5:
        years = years + 1
        Compute interest = value * interest rate
        Add the interest to the value
        Display the value

Finally, we are at the point where we can translate pretty directly into proper programming-language syntax. We still have to choose names for the variables, decide exactly what we want to say to the user, and so forth. Having done this, we could express our algorithm in Java as:

    double principal, rate, interest;  // declare the variables
    int years;
    System.out.print("Type initial investment: ");
    principal = TextIO.getlnDouble();
    System.out.print("Type interest rate: ");
    rate = TextIO.getlnDouble();
    years = 0;
    while (years < 5) {
       years = years + 1;
       interest = principal * rate;
       principal = principal + interest;
       System.out.println(principal);
    }

This still needs to be wrapped inside a complete program, it still needs to be commented, and it really needs to print out more information in a nicer format for the user. But it's essentially the same program as the one in the previous section. (Note that the pseudocode algorithm used indentation to show which statements are inside the loop. In Java, indentation is completely ignored by the computer, so you need a pair of braces to tell the computer which statements are in the loop. If you leave out the braces, the only statement inside the loop would be "years = years + 1;". The other statements would only be executed once, after the loop ends. The nasty thing is that the computer won't notice this error for you, like it would if you left out the parentheses around "(years < 5)". The parentheses are required by the syntax of the while statement. The braces are only required semantically. The computer can recognize syntax errors but not semantic errors.)

One thing you should have noticed here is that my original specification of the problem -- "Compute and display the value of an investment for each of the next five years" -- was far from being complete. Before you start writing a program, you should make sure you have a complete specification of exactly what the program is supposed to do. In particular, you need to know what information the program is going to input and output and what computation it is going to perform. Here is what a reasonably complete specification of the problem might look like in this example:

"Write a program that will compute and display the value of an investment for each of the next five years. Each year, interest is added to the value. The interest is computed by multiplying the current value by a fixed interest rate. Assume that the initial value and the rate of interest are to be input by the user when the program is run."

### 3.2.2 The 3N+1 Problem

Let's do another example, working this time with a program that you haven't already seen. The assignment here is an abstract mathematical problem that is one of my favorite programming exercises. This time, we'll start with a more complete specification of the task to be performed:

"Given a positive integer, N, define the '3N+1' sequence starting from N as follows: If N is an even number, then divide N by two; but if N is odd, then multiply N by 3 and add 1. Continue to generate numbers in this way until N becomes equal to 1. For example, starting from N = 3, which is odd, we multiply by 3 and add 1, giving N = 3*3+1 = 10. Then, since N is even, we divide by 2, giving N = 10/2 = 5. We continue in this way, stopping when we reach 1. The complete sequence is: 3, 10, 5, 16, 8, 4, 2, 1.

"Write a program that will read a positive integer from the user and will print out the 3N+1 sequence starting from that integer. The program should also count and print out the number of terms in the sequence."

A general outline of the algorithm for the program we want is:

       Get a positive integer N from the user.
       Compute, print, and count each number in the sequence.
       Output the number of terms.

The bulk of the program is in the second step. We'll need a loop, since we want to keep computing numbers until we get 1. To put this in terms appropriate for a while loop, we need to know when to continue the loop rather than when to stop it: We want to continue as long as the number is not 1. So, we can expand our pseudocode algorithm to:

    Get a positive integer N from the user;
    while N is not 1:
        Compute N = next term;
        Output N;
        Count this term;
    Output the number of terms;

In order to compute the next term, the computer must take different actions depending on whether N is even or odd. We need an if statement to decide between the two cases:

    Get a positive integer N from the user;
    while N is not 1:
        if N is even:
           Compute N = N/2;
        else
           Compute N = 3 * N + 1;
        Output N;
        Count this term;
    Output the number of terms;

We are almost there. The one problem that remains is counting. Counting means that you start with zero, and every time you have something to count, you add one. We need a variable to do the counting. The variable must be set to zero once, before the loop starts, and it must be incremented within the loop. (Again, this is a common pattern that you should expect to see over and over.) With the counter added, we get:

    Get a positive integer N from the user;
    Let counter = 0;
    while N is not 1:
        if N is even:
           Compute N = N/2;
        else
           Compute N = 3 * N + 1;
        Output N;
        Add 1 to counter;
    Output the counter;

We still have to worry about the very first step. How can we get a positive integer from the user? If we just read in a number, it's possible that the user might type in a negative number or zero. If you follow what happens when the value of N is negative or zero, you'll see that the program will go on forever, since the value of N will never become equal to 1. This is bad. In this case, the problem is probably no big deal, but in general you should try to write programs that are foolproof. One way to fix this is to keep reading in numbers until the user types in a positive number:

    Ask user to input a positive number;
    Let N be the user's response;
    while N is not positive:
       Print an error message;
       Read another value for N;
    Let counter = 0;
    while N is not 1:
        if N is even:
           Compute N = N/2;
        else
           Compute N = 3 * N + 1;
        Output N;
        Add 1 to counter;
    Output the counter;

The first while loop will end only when N is a positive number, as required. (A common beginning programmer's error is to use an if statement instead of a while statement here: "If N is not positive, ask the user to input another value." The problem arises if the second number input by the user is also non-positive. The if statement is only executed once, so the second input number is never tested, and the program proceeds into an infinite loop. With the while loop, after the second number is input, the computer jumps back to the beginning of the loop and tests whether the second number is positive. If not, it asks the user for a third number, and it will continue asking for numbers until the user enters an acceptable input. After the while loop ends, we can be absolutely sure that N is a positive number.)

Here is a Java program implementing this algorithm. It uses the operators <= to mean "is less than or equal to" and != to mean "is not equal to." To test whether N is even, it uses "N % 2 == 0". All the operators used here were discussed in Section 2.5.

    /**
     * This program prints out a 3N+1 sequence starting from a positive
     * integer specified by the user.  It also counts the number of
     * terms in the sequence, and prints out that number.
     */
     public class ThreeN1 {

          public static void main(String[] args) {

             int N;       // for computing terms in the sequence
             int counter; // for counting the terms

             System.out.print("Starting point for sequence: ");
             N = TextIO.getlnInt();
             while (N <= 0) {
                System.out.print(
                       "The starting point must be positive. Please try again: " );
                N = TextIO.getlnInt();
             }
             // At this point, we know that N > 0

             counter = 0;
             while (N != 1) {
                 if (N % 2 == 0)
                    N = N / 2;
                 else
                    N = 3 * N + 1;
                 System.out.println(N);
                 counter = counter + 1;
             }

             System.out.println();
             System.out.print("There were ");
             System.out.print(counter);
             System.out.println(" terms in the sequence.");

         }  // end of main()

     }  // end of class ThreeN1

Two final notes on this program: First, you might have noticed that the first term of the sequence -- the value of N input by the user -- is not printed or counted by this program. Is this an error? It's hard to say. Was the specification of the program careful enough to decide? This is the type of thing that might send you back to the boss/professor for clarification. The problem (if it is one!) can be fixed easily enough. Just replace the line "counter = 0" before the while loop with the two lines:

    System.out.println(N);   // print out initial term
    counter = 1;       // and count it

Second, there is the question of why this problem might be interesting. Well, it's interesting to mathematicians and computer scientists because of a simple question about the problem that they haven't been able to answer: Will the process of computing the 3N+1 sequence finish after a finite number of steps for all possible starting values of N? Although individual sequences are easy to compute, no one has been able to answer the general question. To put this another way, no one knows whether the process of computing 3N+1 sequences can properly be called an algorithm, since an algorithm is required to terminate after a finite number of steps! (Note: This discussion really applies to integers, not to values of type int! That is, it assumes that the value of N can take on arbitrarily large integer values, which is not true for a variable of type int in a Java program. When the value of N in the program becomes too large to be represented as a 32-bit int, the values output by the program are no longer mathematically correct. So the Java program does not compute the correct 3N+1 sequence if N becomes too large. See Exercise 8.2.)

### 3.2.3 Coding, Testing, Debugging

It would be nice if, having developed an algorithm for your program, you could relax, press a button, and get a perfectly working program. Unfortunately, the process of turning an algorithm into Java source code doesn't always go smoothly. And when you do get to the stage of a working program, it's often only working in the sense that it does something. Unfortunately not what you want it to do.

After program design comes coding: translating the design into a program written in Java or some other language. Usually, no matter how careful you are, a few syntax errors will creep in from somewhere, and the Java compiler will reject your program with some kind of error message. Unfortunately, while a compiler will always detect syntax errors, it's not very good about telling you exactly what's wrong. Sometimes, it's not even good about telling you where the real error is. A spelling error or missing "{" on line 45 might cause the compiler to choke on line 105. You can avoid lots of errors by making sure that you really understand the syntax rules of the language and by following some basic programming guidelines. For example, I never type a "{" without typing the matching "}". Then I go back and fill in the statements between the braces. A missing or extra brace can be one of the hardest errors to find in a large program. Always, always indent your program nicely. If you change the program, change the indentation to match. It's worth the trouble. Use a consistent naming scheme, so you don't have to struggle to remember whether you called that variable interestrate or interestRate. In general, when the compiler gives multiple error messages, don't try to fix the second error message from the compiler until you've fixed the first one. Once the compiler hits an error in your program, it can get confused, and the rest of the error messages might just be guesses. Maybe the best advice is: Take the time to understand the error before you try to fix it. Programming is not an experimental science.

When your program compiles without error, you are still not done. You have to test the program to make sure it works correctly. Remember that the goal is not to get the right output for the two sample inputs that the professor gave in class. The goal is a program that will work correctly for all reasonable inputs. Ideally, when faced with an unreasonable input, it should respond by gently chiding the user rather than by crashing. Test your program on a wide variety of inputs. Try to find a set of inputs that will test the full range of functionality that you've coded into your program. As you begin writing larger programs, write them in stages and test each stage along the way. You might even have to write some extra code to do the testing -- for example to call a subroutine that you've just written. You don't want to be faced, if you can avoid it, with 500 newly written lines of code that have an error in there somewhere.

The point of testing is to find bugs -- semantic errors that show up as incorrect behavior rather than as compilation errors. And the sad fact is that you will probably find them. Again, you can minimize bugs by careful design and careful coding, but no one has found a way to avoid them altogether. Once you've detected a bug, it's time for debugging. You have to track down the cause of the bug in the program's source code and eliminate it. Debugging is a skill that, like other aspects of programming, requires practice to master. So don't be afraid of bugs. Learn from them. One essential debugging skill is the ability to read source code -- the ability to put aside preconceptions about what you think it does and to follow it the way the computer does -- mechanically, step-by-step -- to see what it really does. This is hard. I can still remember the time I spent hours looking for a bug only to find that a line of code that I had looked at ten times had a "1" where it should have had an "i", or the time when I wrote a subroutine named WindowClosing which would have done exactly what I wanted except that the computer was looking for windowClosing (with a lower case "w"). Sometimes it can help to have someone who doesn't share your preconceptions look at your code.

Often, it's a problem just to find the part of the program that contains the error. Most programming environments come with a debugger, which is a program that can help you find bugs. Typically, your program can be run under the control of the debugger. The debugger allows you to set "breakpoints" in your program. A breakpoint is a point in the program where the debugger will pause the program so you can look at the values of the program's variables. The idea is to track down exactly when things start to go wrong during the program's execution. The debugger will also let you execute your program one line at a time, so that you can watch what happens in detail once you know the general area in the program where the bug is lurking.

I will confess that I only occasionally use debuggers myself. A more traditional approach to debugging is to insert debugging statements into your program. These are output statements that print out information about the state of the program. Typically, a debugging statement would say something like

    System.out.println("At start of while loop, N = " + N);

You need to be able to tell from the output where in your program the output is coming from, and you want to know the value of important variables. Sometimes, you will find that the computer isn't even getting to a part of the program that you think it should be executing. Remember that the goal is to find the first point in the program where the state is not what you expect it to be. That's where the bug is.

And finally, remember the golden rule of debugging: If you are absolutely sure that everything in your program is right, and if it still doesn't work, then one of the things that you are absolutely sure of is wrong.

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