PWC 388 Dyck Words: I Refuse to Indulge in the Obvious Jokes

When I first glimpsed the title of this week's coding challenge, I knew I had to take precautions against falling into the same humorous pitfalls as others. After a quick Google search, I discovered that the name "Dyck" is pronounced "deek," and as someone with a German-sounding last name, I didn't want to risk any mispronunciations.

Meanwhile, I also remembered the popular Letterkenny catchphrase "Dycks," which brought back memories of a fourteen-year-old's immature humor. With that in mind, I decided it was best to pause and reflect before diving into the task at hand.

Task 1: Dyck Words

The challenge requires generating all valid Dyck words of length 2n, where n is a given integer. A Dyck word is a string consisting of n 'U' (Up) characters and n 'D' (Down) characters, arranged in such a way that no initial prefix of the string has more 'D's than 'U's. The words must be returned as a sorted list in lexicographical order.

Here's a step-by-step approach to solve this challenge:

1. Define a function called task that takes an integer $n as input. This function will handle the special case of $n = 0 and call the recursive function generateDyck to generate the Dyck words.

2. Inside the task function, call generateDyck with the input $n and store the result in a variable called $result. The generateDyck function will return a list of all valid Dyck words of length 2n.

3. To ensure the words are in lexicographical order, reverse the order of the $result list before returning it.

4. Create the generateDyck function that takes the following parameters:

- $n: The input value representing half the length of the Dyck word.

- $dyck: An array reference to store the current Dyck word being constructed (default is an empty array).

- $word: A string representing the current Dyck word (default is 'U').

- $nU: The count of 'U' characters in the current Dyck word (default is 1).

- $nD: The count of 'D' characters in the current Dyck word (default is 0).

- $depth: The current depth of recursion (default is 1).

5. Inside the generateDyck function, use default parameter values to handle the initial call from the task function.

6. The list of generated Dyck words will be accumulated in the $dyck array reference.

7. The recursive function will build the Dyck words by adding 'U' or 'D' characters at a time, following the rules that no initial prefix can have more 'D's than 'U's.

8. The recursion continues until all 'U' and 'D' characters are used up (i.e., $nU == $n).

9. If the current Dyck word is complete (all 'U' and 'D' characters used), add it to the $dyck array reference.

10. Finally, return the $dyck array reference containing all the valid Dyck words in lexicographical order.

By following this approach and leveraging a recursive algorithm, we can efficiently generate all valid Dyck words of length 2n and return them in the required sorted order.