1096. Brace Expansion II

Problem Description

This problem involves interpreting a custom grammar to generate a set of words that an input expression represents. This grammar operates on lowercase English letters and curly braces {}.

The rules are as follows:

1. A single letter like a represents the set {a}.
2. A comma-separated list within curly braces like {a,b,c} creates the union of the single letters, resulting in {a, b, c}.
3. Concatenation of sets means forming all possible combinations of words where the first part comes from the first set and the second part from the second, like ab + cd will give {abcd}.

The task is, given a string expression that uses this grammar, to return a sorted list of all the unique words that it represents.

Intuition

The intuition of the solution relies on the principles of Depth-First Search (DFS). We want to explore all possible combinations of the strings as described by the expression. The idea behind this code is to solve the expression recursively by expanding the braces from the innermost to the outermost.

The process can be broken down into the following steps:

1. Locate the innermost right brace '}' and find the corresponding left brace '{' that starts the current scope.
2. Split the content within the braces by commas to obtain individual segments that need to be combined.
3. Build new expressions by replacing the content within the current braces with each of the segments and recursively process these new expressions.
4. As we're using a set s to store the results, it will automatically handle duplicates and give us a unique set of words.
5. Once all recursive calls are completed, return the words in sorted order as required.

This approach follows a divide-and-conquer strategy, breaking down the expression into smaller parts until it can't be divided anymore, i.e., when no more braces are found. At this point, the simplest expressions are single words that are added to the set s. As the recursion unwinds, the expressions are built up and added to s until the original expression has been fully expanded.

Learn more about Stack, Breadth-First Search and Backtracking patterns.

Solution Approach

The solution algorithm is designed using a recursive function dfs. The idea of this function is to expand the input expression by replacing the expressions within the braces step by step until no more braces exist.

Here's how the algorithm and its components work:

• Data Structure Used: A set s is used to store unique combinations of the expressions as they're being expanded. set in Python takes care of both storing the elements and ensuring all elements are unique.

• Recursive Function dfs: The function dfs is defined which will do most of the heavy lifting. It takes a string exp (partial expression) as its argument.

• Base Case: First, it checks whether there is a '}' in the current expression. If there isn't, it means the expression is fully expanded (i.e., has no more braces to process), and it can be added directly to the set s.

• Finding Braces Pairs: If a '}' is found, it locates the left brace '{' that matches with this right brace by using exp.rfind('{', 0, j - 1), which searches for the last '{' before the found '}'.

• Processing Innermost Expression:

  • It splits the content located between these braces on commas, resulting in a list of smaller expressions that are not yet combined.
  • It generates new partial expressions by taking the part of the expression before the found '{' (a), each of the smaller expressions (b), and the part after the '}' (c) and calls dfs(a + b + c) on them.

• Recursion: This recursive call continues to expand more braces within the new partial expressions until all possibilities are added to the set s.

• Sorting: Once the recursion is completed, the function braceExpansionII returns sorted(s), which is a list of all the unique words represented by the original expression, sorted in lexicographical order.

To summarize, the algorithm uses DFS with recursion to expand the expressions by iteratively replacing the curly braces with their possible combinations, collecting results in a set to maintain uniqueness, and then returning the sorted list of those results.

Example Walkthrough

Let's say we have the following input expression:

1"{a,b}{c,d}"

According to the problem description and the rules of the given grammar, this expression could generate the following set of words: {ac, ad, bc, bd}.

Here's a step-by-step illustration of the solution approach using this example:

1. Start DFS: The expression has braces, so our dfs function starts processing it.

2. Locate Braces: The dfs function finds the rightmost '}' at index 7 and then finds its corresponding left brace '{' at index 4.

3. Split Content: The content within these braces is "c,d". The dfs function splits this into ["c", "d"].

4. Build Expressions: The expression before the '{' is {a,b}, and there's nothing after '}'. So we create new expressions by replacing {c,d} with c and with d from the split results. The new partial expressions are {a,b}c and {a,b}d.

5. Recursion: Now, we recursively call dfs on {a,b}c and {a,b}d:

   • For {a,b}c:

     • Find '}' at index 3 and corresponding '{' at index 0.
     • Split "a,b" into ["a", "b"].
     • Create new expressions ac and bc.
     • As these new expressions contain no braces, add them directly to the set s.

   • For {a,b}d:

     • Repeat the same process as for {a,b}c.
     • Split "a,b" into ["a", "b"].
     • Create new expressions ad and bd.
     • Add these to the set s, too.

6. Completion of DFS: The set s now contains {"ac", "bc", "ad", "bd"}.

7. Return Sorted Result: Finally, once all recursion is done, sorted(s) returns:

1["ac", "ad", "bc", "bd"]

This is the sorted list of all unique words represented by the input expression.

By following the given solution approach, we can see how the problem of expanding and combining expressions is broken down into manageable steps using DFS and recursion. Each level of recursion deals with a simpler expression, and the set s ensures that all computed words are unique and stored efficiently. Once all possibilities are explored, the sorted content of this set is the answer.

Solution Implementation

Time and Space Complexity

Time Complexity

The provided code performs a depth-first search (DFS) to generate all possible combinations of strings that can result from the given expression, following the rules of brace expansion.

The time complexity of the algorithm is hard to analyze precisely without more context regarding the nature of expression. However, given that the code involves recursive calls for every substring result, we can make some general observations:

• Each recursive call is made after finding a pair of braces {} and for each element in the comma-separated list inside.
• For every split by comma inside the braces, a new recursive DFS call is made, therefore, in the worst case, the number of recursive calls for each level of recursion is O(n), where n is the size of the split list.
• The maximum depth of recursion is related to the number of nested brace pairs. If we assume the maximum number of nested brace pairs is k, then to some degree we can say the depth of recursion is O(k).
• The cost of string concatenation in Python is O(m), where m is the length of the resulting string.

Considering these factors, in the worst-case scenario where the recursion is deep and the number of elements to combine at each level is large, the time complexity could grow exponentially with respect to the number of brace pairs and the number of individual elements within them. Thus, it can be considered O(n^k) or worse. This is a rough estimate as the actual time complexity can vary dramatically with the structure of the expression.

Space Complexity

The space complexity of the algorithm is also influenced by several factors:

• The recursion depth k contributes to the space complexity due to call stack usage. Hence, the implicit stack space is O(k).
• The set s stores all unique combinations which can be at most O(2^n) where n is the number of elements in expression, since each element inside a brace can either be included or excluded.
• Each recursive call involves creating new strings, and if memory allocated for these strings is not efficiently handled by the Python's memory management, it could add to the space complexity.

Taking these into account, the worst-case space complexity can be considered O(2^n + k) for storing combinations and stack space, but similar to the time complexity, this heavily depends on the input expression structure.

Learn more about how to find time and space complexity quickly using problem constraints.