Beginners often have trouble figuring out how exactly recursive functions work. Today, you will be working with fairly short recursive functions and try to investigate the magic of recursion. It is not really magic once you understand how it works, but it is not a bad idea to think of it as magic at the beginning - you can certainly do some pretty neat tricks with it.
The main idea of recursion is to solve a large problem by reducing it to smaller and smaller version of it and then finally solving a trivial problem.
In the example below, the objective is to calculate factorial of a number,
\[n! = \begin{cases} n \times (n-1)! & \mbox{if } n > 0 \\ 1 & \mbox{if } n = 0 \end{cases}\]The first person does not know the answer to $10!$ right away. But they do know how to reduce the problem so that it is a bit smaller, $9!$. They also know how to use the solution to the smaller problem to calculate the solution for the larger problem, $10\times 9!$. Each person along the chain makes the problem smaller and smaller and smaller, until the final person can answer the question directly without asking anybody else for help, $0!$.
In a recursive function, we make the problem smaller by making a function call to the recursive function with different arguments (such that those arguments make the problem a little bit easier).
The question that is so simple that it does not need further recursive calls is called the base case - this is when we can give the answer directly.
Here is the same problem solved using a Java function:
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//assume n >= 0
long factorial ( int n ) {
if (n == 0) //base case
return 1;
else //recursive case
return n * factorial(n-1);
}
You'll be working with the CodingBat exercises again.
Let's see if you can solve some of the problems related to recursion.
The website tests your code using many inputs. Try to write code solutions to the following problems.
There are many more problems in Recursion 1 section (http://codingbat.com/java/Recursion-1) - use them for extra practice.
To get progress and participation credit for this lab, take a screenshot of one or more of your submitted problems showing the test results and submit the screenshot(s) to Gradescope under Lab 6.
You can use PythonTutor website to visualize code execution in several different languages. (Python in its name may be miss-leading. It works with many programming languages.)
Follow the link to the two recursive functions on PythonTutor website: https://goo.gl/Wfyp4C (it tends to be slow to start, so be patient; if it fails, just try again by clicking Visualize Execution; if it says there are errors, just try it again).
Use the Forward button to execute the code one line at a time. You can also use the Back button to go backwards (this is something that does not work in a debugger, unfortunately, but for good reasons).
As you are stepping through the code, there are boxes that appear on the right hand side. Those boxes symbolize the stack frames created for each function on the program's stack as the program is running. They contain storage for all of the local variables and parameters. They also contain additional information that allows the program to "jump" from one place in the code to the next.
Investigate the two functions listed in the example and answer the questions
fun3?fun3 (each call has its own stack frame)?
List the value of the parameter together with the return value.fun3 do?fun4.
What does this function do? (answer the same questions as for fun3)These are problems that should be solved recursively, but with slightly more sophisticated recursion than in the first set of problems. Many of them require recursive backtracking technique that we discussed in class.
Keep in mind that the main idea of recursion is to solve a large problem by reducing it to smaller and smaller version of it and then finally solving a trivial problem. Before writing code, spend some time thinking about what the simple cases are (i.e., base case) and how to move from a large problem to a smaller problem (recursive case).
Here are some problems that you should look at. Try to solve all of the problems.
There are several more problems in Recursion 2 section (https://codingbat.com/java/Recursion-2 ) - use them for practice on your own.