Solution Explanation

The problem asks to count the number of vowel substrings within a given string that contain all five vowels (a, e, i, o, u). The solutions presented use different approaches to achieve this.

Solution 1 (More Efficient): This solution iterates through all possible substrings and checks if they satisfy the conditions.

• Outer Loop: The outer loop iterates through each starting index i of the substring.
• Inner Loop: The inner loop iterates through each ending index j of the substring starting from i.
• Set Check: A set (or HashSet in Java, unordered_set in C++) is used to efficiently track the unique vowels present in the current substring. The set only contains vowels encountered so far in the current substring. If the size of the set becomes 5, it means all five vowels are present, and the counter ans is incremented.
• Optimization: The inner loop breaks if a non-vowel character is encountered, avoiding unnecessary iterations.

Solution 2 (Slightly Less Efficient): This solution is similar but uses a slightly different approach to check for vowel substring.

• Outer Loop: The outer loop iterates through each starting index i.
• Inner Loop: The inner loop iterates through characters from index i to the end of the string.
• Set Check: Similar to Solution 1, a set is used to track unique vowels. If the set contains all five vowels, ans is incremented.
• Optimization: The inner loop breaks if a non-vowel character is encountered.

Time Complexity Analysis:

Both solutions have a time complexity of O(n^2), where n is the length of the input string. The nested loops dominate the runtime. In the worst-case scenario (a string with many vowels), the inner loop will iterate almost to the end of the string for many iterations of the outer loop.

Space Complexity Analysis:

The space complexity is O(1) for both solutions. The space used by the set (or equivalent data structure) remains constant, regardless of the input string length. The size of the set is always bounded by the number of vowels (5).

Code Examples (Python, Java, C++, Go, TypeScript):

The provided code examples demonstrate the implementation of both solutions in various programming languages. They follow the algorithmic descriptions outlined above. Note that Solution 1 is generally preferred due to its slightly more efficient handling of substrings, especially for larger input strings. The difference may not be noticeable for smaller inputs.

Example (Python - Solution 1):

class Solution:
    def countVowelSubstrings(self, word: str) -> int:
        n = len(word)
        s = set('aeiou')
        return sum(set(word[i:j]) == s for i in range(n) for j in range(i + 1, n + 1))

This concise Python code leverages generator expressions for a compact and efficient implementation of Solution 1.

The other languages' code examples are similarly straightforward implementations of the described algorithms. They all share the same core logic but adapt to the specific syntax and data structures of each language.