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# Sample code to read input and write output:
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NAME = input() # Read input from STDIN
// ... rest of solution available after purchaseExplanation
To approach this problem, it's essential to understand that it essentially asks us to find all possible ways to segment a given string into valid words from a provided dictionary. The solution revolves around techniques related to string segmentation, backtracking, and dynamic programming. Here’s a comprehensive breakdown of how to think through the problem.
1. Understand the problem requirements:
- You have a string A with no spaces.
- You have a dictionary of unique words.
- You need to split A into a sequence of words, where each word is from the dictionary.
- The entire string must be used; no characters are left out.
- Words can be reused multiple times.
- You must find all feasible sentences formed by inserting spaces accordingly.
2. Recognize the problem type:
- This is a classic word break problem variant, but instead of checking feasibility (i.e., can the string be segmented?), you have to output all possible valid segmentations.
- The challenge is combinatorial since multiple segmentations may exist.
3. Naive approach - Backtracking:
- Start from the first character of the string.
- Try every possible substring from the current position to the end.
- Check if the substring exists in the dictionary.
- If yes, recursively solve the problem for the remainder of the string.
- Collect all valid sentences formed by combining the current substring with solutions from the recursive calls.
- This approach is straightforward but can be exponentially expensive due to repeated subproblems.
4. Improving efficiency - Use memoization:
- To avoid repeated calculations on the same substring, use a memoization cache.
- Store the results (all possible sentences) for each starting index of the string once computed.
- If you encounter the same index again, return the cached results instead of recomputing.
- This optimization converts the exponential time backtracking into a more manageable time complexity, though it still depends on the number of valid segmentations.
5. Dictionary lookups:
- Efficiently checking whether a substring is in the dictionary is crucial.
- Store all dictionary words in a hash set or trie data structure for quick lookup.
- When iterating through substrings, check membership efficiently.
6. Building the result sentences:
- As you find valid words from the current index, combine them with sentences formed from the remaining string.
- At each recursive step, maintain partial solutions as strings.
- For example, if the substring is a valid word and the remainder forms sentences S, the current sentences will be word + " " + s for each s in S.
- Handle the base case carefully: when you reach the end of the string, return an empty string or list containing an empty string to signify completion.
7. Managing output:
- Since multiple sentences may exist, store them in a list.
- The final output is the aggregation of all valid sentences constructed from the entire string.
8. Edge cases and constraints:
- Empty string or dictionary: If the string is empty, return an empty list or handle as a base case.
- Words that are prefixes of other words: Ensure all possible splits are tried.
- Large inputs: Memoization helps but the number of valid segmentations can still be very large.
- Words with overlapping segments: Backtracking ensures all possible combinations are explored.
9. High-level algorithm summary:
- Preprocess dictionary for quick membership checking.
- Use a recursive function that, given a start index, returns all valid sentences from that point forward.
- At each index, iterate through possible end indices to extract candidate substrings.
- If a candidate is in the dictionary, recursively process the remainder of the string.
- Use memoization to store and reuse results.
- Concatenate words and sentences to build complete results.
10. Practical considerations:
- Since the number of valid sentences could be very large, think about limiting output or handling memory constraints.
- The order of output can be based on the order of exploration; typically, this is acceptable unless sorting is required.
- Testing with small and large inputs helps verify correctness and efficiency.
By thinking through these points and combining recursive backtracking with memoization, you can systematically explore all feasible sentence formations that satisfy the given constraints. This structured approach ensures correctness while managing complexity.