In other words, it is the mechanism that stops this process of ever more recursive calls and an ever growing stack of function calls waiting on the return of other function calls. Therefore, we should make sure that every function call eventually hits the base case in order to avoid infinite recursion. Once we hit it, we can start taking function calls off the top of the stack and returning from them. Base Cases Whether or not we are in the base case is determined by the parameters we're passed.
Specify the base case of this function. Answer: When k is zero. Does every function call eventually hit the base case? Answer: Yes. No matter how big the number in our initial call to sum , we must eventually reach our base case, because the argument to our recursive calls keep getting smaller and smaller, until they finally reach zero.
Does every function call eventually hit the base case , really? And for every possible door you can move through, the computer remembers those what ifs, and for every door after that, and after that, etc, until the end is found. Answer: That was a trick question There is no base case in the code. You need to check at the start if the room is the exit. If it is, no recursion! Answer: There are various techniques. If the room is a structure or object you can add the visited field direction to the room.
Answer: The answer to that can be found by thinking about the following question: What would happen if the maze was a giant grid of identically sized rectangular rooms, each with doors on every wall? Imagine you go North through the first door, then East through the next rooms door, then South through that rooms door, and then West through that rooms door. Where do you end up? Back where you started! Worse, you might continue to make this loop for ever.
How would a intrepid adventurer solve this problem? One answer to that is by using a piece of chalk and putting a big X on the floor of every room you enter. Thus when you come back to a room with an X on the floor, you know you needn't enter. In the case of the program, a boolean flag "seen" or "visited" should be used.
Every room has a flag. Every room starts with the flag being set to false. When you visit a room, you set the flag to true. Finally, in the "base case" you have a line such as:.
Some Computer related examples include: Adding a list of numbers, Computing the Fibonacci sequence, computing a Factorial, and Sudoku. Question: What is a recursive solution to summing up a list of numbers? Lets write a recursive function to compute the Nth number in the Fibonacci sequence. The problem here is the computer must re-evaluate the same Fibonacci number over and over again. So when it comes time to evaluate fib 6 , we are going to evaluate fib 5 over again, and then evaluate fib 4 even more times.
Note: The algorithm requires the use of a "merge function" which merges two ordered lists back into a single ordered list! Take the first letter of the word. We can form one set of subsequences that include that letter, and another set of subsequences that exclude that letter, and those two sets completely cover the set of possible subsequences.
Base cases often correspond to emptiness — the empty string, the empty list, the empty set, the empty tree, zero, etc. If every recursive step shrinks the problem, and the base case lies at the bottom, then the recursion is guaranteed to be finite. A recursive implementation may have more than one base case, or more than one recursive step. Recursive methods have a base case and a recursive step. What other concepts from computer science also have the equivalent of a base case and a recursive step?
The recursive implementation we just saw for subsequences is one possible recursive decomposition of the problem. We took a solution to a subproblem — the subsequences of the remainder of the string after removing the first character — and used it to construct solutions to the original problem, by taking each subsequence and adding the first character or omitting it.
This is in a sense a direct recursive implementation, where we are using the existing specification of the recursive method to solve the subproblems. In this case, what if we built up a partial subsequence using the initial letters of the word, and used the recursive calls to complete that partial subsequence using the remaining letters of the word? This subsequencesAfter method is called a helper method. It satisfies a different spec from the original subsequences , because it has a new parameter partialSubsequence.
This parameter fills a similar role that a local variable would in an iterative implementation. It holds temporary state during the evolution of the computation. The recursive calls steadily extend this partial subsequence, selecting or ignoring each letter in the word, until finally reaching the end of the word the base case , at which point the partial subsequence is returned as the only result.
Then the recursion backtracks and fills in other possible subsequences. To finish the implementation, we need to implement the original subsequences spec, which gets the ball rolling by calling the helper method with an initial value for the partial subsequence parameter:. Your decision to decompose the recursion this way instead of another way is entirely implementation-specific.
That exposes your implementation to the client and reduces your ability to change it in the future. Use a private helper function for the recursion, and have your public method call it with the correct initializations, as shown above. Here is his implementation:. Suppose we call subsequencesLouis "c" followed by subsequencesLouis "a". Alyssa P. Hacker throws up her hands when she sees what Louis did.
Which of these statements are true about his code? Unfortunately a static variable is simply a bad idea in recursion. You can step through an interactive visualization of this version to see what happens. It will produce these recursive calls to subsequencesLouis :. When each of these calls starts , what is the value of the static variable partialSubsequence? Suppose we want to convert an integer to a string representation with a given base, following this spec:. For example, stringValue 16, 10 should return "16" , and stringValue 16, 2 should return "".
One recursive step here is straightforward: we can handle negative integers simply by recursively calling for the representation of the corresponding positive integer:. This shows that the recursive subproblem can be smaller or simpler in more subtle ways than just the value of a numeric parameter or the size of a string or list parameter. We have still effectively reduced the problem by reducing it to positive integers. Thinking about the number as we would write it down on paper, we could either start with 8 the leftmost or highest-order digit , or 9 the rightmost, lower-order digit.
Instead, a better way to decompose n is to take its remainder modulo base which gives the rightmost digit and also divide by base which gives the subproblem, the remaining higher-order digits :. Think about several ways to break down the problem, and try to write the recursive steps. You want to find the one that produces the simplest, most natural recursive step.
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