lab_quacks

Assignment Description

In this lab, you will learn to think recursively and apply it to the stack and queue data structures. You might also practice templates.

Lab Insight

Stacks and queues are incredible data structures used in a wide range of applications throughout the world. You have already seen a great example of a queue in CS 225. We use the queue data structure to help us manage our office hours. Queues can do so many incredible things like helping schedule tasks for a computer or managing user priority based on a FIFO (first in, first out) principle. You will find stacks useful in classes like CS 438, Communication Networks. You will find queues useful in CS 241, Systems Programming. In general, these data structures are very versatile and useful throughtout the software world.

Recursion

What is recursion? Recursion is a way of thinking about problems that allows the computer to do more of the heavy lifting for us. It is analogous to the mathematical definition of recursive functions, where you can define a function call in terms of calls to itself and basic arithmetic operations, but not in terms of loops.

Why recursion? While being able to think recursively is one of the harder parts of computer science, it is also one of the most powerful. In fact, there are whole languages that entirely use recursion instead of loops, which, even though it may seem inefficient, leads to some very useful optimizations a compiler can make when dealing with such code. There are probably more problems in computer science that are simpler recursively than they are iteratively (using loops). Also, once you have a recursive algorithm, it is always possible to transform it into an iterative algorithm using a stack and a while loop. In this way, computer scientists can think about problems recursively, then use that recursive solution to make a fast iterative algorithm (and in the grand scheme of big-O notation, using recursion has little overhead compared to the rest of the running time). Here we’ll only ask you to do the first part.

How do I write recursively? Recursion just means calling a function within itself. This may sound crazy, but in fact it is not. Let’s take an iterative function to calculate the factorial of a number \(n\), \(n!\):

int factorial(int n)
{
    int result = 1;
    for (int i = 1; i <= n; i++)
        result = result * i;
    return result;
}

Okay, so four lines of code. Pretty short and understandable. Now let’s look at a recursive version:

int factorial(int n)
{
    if (n == 0) return 1;
    return (factorial(n-1) * n);
}

Only two lines of code! (Depending on whether you like putting your return statement on the same line.) Even on such a small problem, recursion helps us express ourselves more concisely. This definition also fits better with the mathematical definition:

A typical recursive function call consists of three parts. Let’s examine the function more closely to see them. Here’s the same code again, with more discussion.

int factorial(int n)
{
    if (n == 0)    // Here is our base case.
        return 1;  // The base case is the smallest problem we can think of,
                   // one we know the answer to. This is the "n = 0" case in
                   // the mathematical definition.

    // optional 'else' here

    return (factorial(n-1) // This is our recursive step. Here we are solving a
                           //   smaller version of the same problem. We have to
                           //   make a leap of faith here - trust that our
                           //   solution to the (n-1) case is correct.  This is
                           //   the same as the mathematical definition, where
                           //   to figure out n!, we need to first figure out
                           //   (n-1)!
               * n         // Here is our incremental step. We are transforming
                           //   our solution to the smaller problem into the
                           //   solution to our larger problem. This is the
                           //   same * n from the mathematical definition.
           );
}

Checking Out the Code

After reading this lab specification, the first task is to check out the provided code from the class repository.

To check out your files for the third lab, run the following command in your cs225git directory:

git fetch release
git merge release/lab_quacks -m "Merging initial lab_quacks files"

This should update your directory to contain a new directory called lab_quacks.

As usual, to see all the required functions, check out the Doxygen.

Recursive Exercises

Sum of Digits

Given a non-negative int n, return the sum of its digits recursively (no loops). Note that modulo (%) by 10 yields the rightmost digit (126 % 10 == 6), while divide (/) by 10 removes the rightmost digit (126 / 10 == 12).

int sumDigits(int n);

sumDigits(126) -> 1 + 2 + 6 -> 9
sumDigits(49)  -> 4 + 9     -> 13
sumDigits(12)  -> 1 + 2     -> 3

Triangle

We have triangle made of blocks. The topmost row has 1 block, the next row down has 2 blocks, the next row has 3 blocks, and so on:

       *        1 block
     *   *      2 blocks
   *   *   *    3 blocks
 *   *   *   *  4 blocks
............... n blocks

Compute recursively (no loops or multiplication) the total number of blocks in such a triangle with the given number of rows.

int triangle(int rows);

triangle(0) -> 0
triangle(1) -> 1
triangle(2) -> 3

The sum Function

Write a function called sum that takes one stack by reference, and returns the sum of all the elements in the stack, leaving the original stack in the same state (unchanged). You may modify the stack, as long as you restore it to its original values. You may use only two local variables of type T in your function. Note that this function is templatized on the stack’s type, so stacks of objects overloading the addition operator (operator+) can be summed. Hint: think recursively!

template <typename T>
T QuackFun::sum(stack<T> & s);

Non Recursive Exercises

Balancing Brackets: the isBalanced Function

For this exercise, you must write a function called isBalanced that takes one argument, an std::queue, and returns whether the string represented by the queue has balanced brackets. The queue may contain any characters, although you should only consider square bracket characters, ‘[’ and ‘]’, when considering whether a string is balanced. To be balanced, a string must not have any unmatched, extra, or hanging brackets. For example, the string [hello][] is balanced, [[][[]a]] is balanced, []] is unbalanced, ][ is unbalanced, and ))))[cs225] is balanced. It’s possible to solve this problem without using a stack, but in the spirit of this lab, you should use one in your solution!

bool isBalanced(queue<char> input);

The scramble Function

Your task is to write a function called scramble that takes one argument: a reference to a std::queue.

template <typename T>
void QuackFun::scramble(queue<T> & q);

You may use whatever local variables you need. The function should reverse the order of SOME of the elements in the queue, and maintain the order of others, according to the following pattern:

• The first element stays on the front of the queue.
• Then the next two elements are reversed.
• Then the next three elements are placed on the queue in their original order.
• Then the next four elements are reversed.
• Then the next five elements are place on the queue in their original order.
• etc.

Hint: You’ll want to make a local stack variable.

For example, given the following queue,

front                                         back
0   1 2   3 4 5   6 7 8 9   10 11 12 13 14   15 16

we get the following result:

front                                         back
0   2 1   3 4 5   9 8 7 6   10 11 12 13 14   16 15

Any “leftover” numbers should be handled as if their block was complete. (See the way 15 and 16 were treated in our example above.)

(Extra-Credit) The verifySame function

Write the recursive function verifySame whose function prototype is below. The function should return true if the parameter stack and queue contain only elements of exactly the same values in exactly the same order, and false otherwise (see example below). You may assume the stack and queue contain the same number of items!

We’re going to constrain your solution so as to make you think hard about solving it elegantly:

• Your function may not use any loops;
• In your function you may only declare ONE local boolean variable to use in your return statement, and you may only declare TWO local variables of parametrized type T to use however you wish. No other local variables can be used; and
• After execution of verifySame, the stack and queue must be unchanged. Be sure to comment your code VERY well.

Stack
+---+
| 1 | top
+---+
| 2 |              Queue
+---+      +---+---+---+---+---+
| 3 |      | 1 | 2 | 3 | 4 | 5 |
+---+      +---+---+---+---+---+
| 4 |      back            front
+---+
| 5 | bottom
+---+

Click here to see the answer

Testing Your Code

Run the Catch tests as follows:

make test
./test

Grading Information

The following files are used in grading:

• exercises.cpp
• exercises.h
• quackfun.cpp
• quackfun.h

All other files including any testing files you have added will not be used for grading.