## 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.

## 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 $n$, $n!$:

Example
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:
Example
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:

$n! = \begin{cases} 1 & \text{if n = 0,} \\ (n-1)!\times n & \text{if n > 0.} \end{cases}$

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.

Example
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.

STL Stack and Queue

These activities use the standard template library’s stack and queue structures. The interfaces of these abstract data types are slightly different than in lecture, so it will be helpful for you to look up “STL Stack” and “STL Queue” on Google (C++ reference has good information). In particular, note that the pop() operations do not return the element removed, and that you must look that up before calling pop().

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

## Recursive Exercises

No loops!

You may not use any loops for this section! Try to think about the problem recursively: in terms of a base case, a smaller problem, and an incremental step to transform the smaller problem to the current problem.

### 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

Note

These examples were stolen from http://codingbat.com/java/Recursion-1. All credit goes to CodingBat. If you are having a hard time with sum (below), we encourage you to go to CodingBat and try more recursive exercises. These are in Java, but there are links at the bottom of the page describing the differences of strings and arrays in Java from C++, which are minor.

### 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!

STL Stack

We are using the Standard Template Library (STL) stack in this problem. Its pop function works a bit differently from the stack we built. Try searching for “STL stack” to learn how to use it.

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.

For this function, you may only create a single local variable of type stack<char>! No other stack or queue local objects may be declared.

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.)

STL Queue

We are using the Standard Template Library (STL) queue in this problem. Its pop function works a bit differently from the queue we built. Try searching for “STL queue” to learn how to use it.

## (Extra-Credit) The verifySame function

Complier errors

Submitting code that doesn’t compile will result in a zero on the entire lab. This includes code in the extra credit portion and should be common sense. Be sure to test your code before you submit.

Extra Credit

This function is NOT part of the standard lab grade, but is extra credit. It was also a previous exam question, and something similar could show up again.

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.
Example

This stack and queue are considered to be the same. Note that we match the bottom of the stack with the front of the queue. No other queue matches this stack.

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


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