This problem involves identifying concatenated words in a list by checking whether a word can be formed by concatenating at least two other words from the list. The solution leverages dynamic programming and trie structures to efficiently determine the valid concatenated words. Techniques like depth-first search and state transition dynamic programming play key roles in solving this problem.
Problem Statement
Given a list of unique strings, return all words in the list that are formed by concatenating at least two other words from the same list. A concatenated word must be formed by joining shorter words from the array, not necessarily distinct.
For example, in the list ["cat", "cats", "catsdogcats", "dog", "dogcatsdog", "hippopotamuses"], the words "catsdogcats", "dogcatsdog", and "ratcatdogcat" are valid concatenated words, while others are not. The challenge is to efficiently identify these concatenated words for large inputs, which requires careful handling of the problem's constraints.
Examples
Example 1
Input: words = ["cat","cats","catsdogcats","dog","dogcatsdog","hippopotamuses","rat","ratcatdogcat"]
Output: ["catsdogcats","dogcatsdog","ratcatdogcat"]
"catsdogcats" can be concatenated by "cats", "dog" and "cats"; "dogcatsdog" can be concatenated by "dog", "cats" and "dog"; "ratcatdogcat" can be concatenated by "rat", "cat", "dog" and "cat".
Example 2
Input: words = ["cat","dog","catdog"]
Output: ["catdog"]
Example details omitted.
Constraints
- 1 <= words.length <= 104
- 1 <= words[i].length <= 30
- words[i] consists of only lowercase English letters.
- All the strings of words are unique.
- 1 <= sum(words[i].length) <= 105
Solution Approach
Dynamic Programming Approach
Use dynamic programming (DP) to break down the problem into smaller subproblems. We create a DP table for each word and check if it can be formed by concatenating smaller words from the list. This solution requires iterating over substrings of a word to see if they exist in the set of words.
Trie Data Structure
A trie is built to store all the words in the list. This allows efficient lookup of substrings within the words. By combining the trie structure with dynamic programming, we can optimize the process of checking whether a word can be split into smaller valid words.
Depth-First Search
A depth-first search (DFS) can be used to traverse through the trie and check if a word can be decomposed into valid substrings that are also words in the list. This method is useful for ensuring that every potential split of a word is explored, making it an essential part of the solution.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | O(M ^ 3 \cdot N) |
| Space | O(N \cdot M) |
The time complexity is O(M ^ 3 * N), where M is the maximum length of a word and N is the number of words. This accounts for checking every possible substring and validating it using dynamic programming or trie-based lookups. The space complexity is O(N * M), where N is the number of words and M is the average length of the words, due to storage of the trie and dynamic programming tables.
What Interviewers Usually Probe
- Check if the candidate understands how to use dynamic programming and recursion together to optimize the search for concatenated words.
- Look for knowledge of optimizing searches using trie data structures and recognizing when to use them for large-scale problems.
- Assess if the candidate can handle word manipulations effectively, considering time and space complexities in relation to the problem's constraints.
Common Pitfalls or Variants
Common pitfalls
- Not using a trie to optimize substring lookups, which can lead to inefficient solutions for large inputs.
- Failing to properly handle words that can overlap, leading to incorrect results.
- Using brute force methods without considering time complexity, which will lead to performance issues when the input list is large.
Follow-up variants
- Implementing this problem using only dynamic programming without a trie.
- Allowing a word to be formed from any subset of words (not just at least two) in the list.
- Handling edge cases where a word might be a concatenation of its own prefix and suffix.
How GhostInterview Helps
- GhostInterview walks you through the essential concepts of dynamic programming and trie-based search, helping you recognize key patterns in the problem.
- By focusing on state transition dynamic programming, GhostInterview enables you to solve similar problems efficiently during interviews.
- GhostInterview helps sharpen your understanding of substring searching and optimizes time and space complexities for real-world scenarios.
Topic Pages
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- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the core approach for solving Concatenated Words?
The core approach is to use dynamic programming, depth-first search, and a trie to break down words and identify those that can be formed by concatenating other words in the list.
How can I optimize the solution for large input sizes?
Using a trie structure for efficient substring lookups and dynamic programming to minimize redundant calculations can help optimize the solution for large inputs.
What is the role of dynamic programming in this problem?
Dynamic programming helps break down the problem into smaller subproblems, checking if a word can be formed by concatenating smaller words from the list without recalculating the same results repeatedly.
Can this problem be solved without using a trie?
Yes, it can be solved using only dynamic programming, but a trie can significantly improve efficiency by enabling fast substring lookups.
What is the time complexity of the optimal solution?
The time complexity of the optimal solution is O(M ^ 3 * N), where M is the maximum word length and N is the number of words.
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