Solution Explanation

This problem asks to generate all possible strings from a given input string containing braced options. The solution uses a backtracking approach to explore all combinations. Let's break down the solution step-by-step:

Approach

1. Parsing the Input String: The input string s needs to be parsed to extract the individual options within the curly braces {}. The convert function (in both Python and Java solutions) handles this. It iteratively searches for { and }. When it finds a pair, it splits the content within the braces by commas, creating a list of options. If there are no braces, it splits the entire string by commas (treating it as a single group of options). The parsed options are stored in the items list (a list of string arrays in Java, and a list of lists in Python).

2. Backtracking (Depth-First Search): The core logic resides in the dfs function. It uses a recursive backtracking approach to generate all possible combinations.

   • Base Case: If the dfs function has processed all groups of options (i == len(items)), it means a complete word has been formed. The word (represented by the t list) is added to the ans list.

   • Recursive Step: For each group of options at index i, the function iterates through every option in that group. It appends the current option to the temporary word (t), makes a recursive call to dfs to explore further options, and then removes the option from t (backtracking) to explore other possibilities within the current group.

3. Sorting and Returning: Finally, the ans list containing all generated words is sorted lexicographically and returned.

Time Complexity Analysis

The time complexity is dominated by the backtracking step. In the worst case, if each group of options has k elements, and there are n groups, the number of possible words is kn. The dfs function explores each of these possibilities, resulting in O(kn) time complexity. The sorting step adds O(kn log kn), which simplifies to O(kn log k) but is still dominated by the backtracking. Therefore, the overall time complexity is O(kn log k), where 'n' is the number of groups of options and 'k' is the maximum number of options within a group.

Space Complexity Analysis

The space complexity is determined by:

• The items list, which stores the parsed options. In the worst-case scenario, the size of items is proportional to the length of the input string s, so it is O(s).
• The ans list, which stores the generated words. In the worst case, the size of ans is kn.
• The temporary list t used during the depth-first search, which has a maximum depth of n. Therefore, the overall space complexity is O(s + kn), where 's' is the length of the input string, 'n' is the number of groups, and 'k' is the maximum options per group. The kn term dominates, making the space complexity essentially O(kn).

Code in Different Languages

The provided solutions already contain Python and Java implementations. Other languages would follow a similar structure, utilizing recursive backtracking and string manipulation. The core algorithm remains consistent.