Understanding the Problem
We are given a list of words that are sorted lexicographically according to the rules of an unknown language. Our task is to determine the correct order of characters in this alien language based on the order of words. The key idea is that the character order must satisfy the relative positions of characters as implied by the word list.
This is a classic case of extracting a topological ordering of characters from the word relationships.
Step-by-Step Solution with Example
step 1: Understand what the input means
Suppose the input is a list of words like [ "baa", "abcd", "abca", "cab", "cad" ]
. These words are sorted in an alien language. From this, we need to find the character order of that alien alphabet.
step 2: Build a graph of characters
Each unique character will be a node in a graph. By comparing adjacent words, we extract ordering rules. For example:
- Compare "baa" and "abcd" → 'b' vs 'a' → since 'b' ≠ 'a', we know 'b' comes before 'a' → add edge b → a.
- Compare "abcd" and "abca" → 'd' vs 'a' → 'd' comes before 'a' → add edge d → a.
- Compare "abca" and "cab" → 'a' vs 'c' → 'a' comes before 'c' → add edge a → c.
- Compare "cab" and "cad" → 'b' vs 'd' → 'b' comes before 'd' → add edge b → d.
step 3: Prepare in-degrees for topological sort
We calculate the in-degree (number of incoming edges) for each node (character). This helps identify nodes with no dependencies — these can appear first in the alien order.
step 4: Topological Sorting using Kahn’s Algorithm
Start with characters having in-degree 0. Add them to a queue and begin removing them one by one, appending to the result string and decreasing the in-degree of their neighbors. If a neighbor's in-degree becomes 0, add it to the queue.
step 5: Check for cycle
If at the end of this process, our result contains fewer characters than expected, then there is a cycle (conflicting character ordering), and we cannot determine a valid order.
step 6: Return the result
The result string formed by topological sort will represent the correct character order in the alien language if no cycle exists.
Edge Cases
- Prefix issue: If a word is a prefix of a longer word and comes later (e.g., “apple” before “app”), this is invalid. We should detect this and return an empty result.
- Disconnected characters: If a character appears in the word list but no ordering info is derived from it, we still need to include it in the result.
- Duplicate words: Should be ignored as they give no additional info.
- Single word input: We can return any order of the unique characters in that word.
Finally
This problem is a combination of graph building and topological sorting. By carefully comparing adjacent words and constructing a graph of character dependencies, we can extract a valid character order for the alien language. Kahn’s algorithm is particularly useful because it also helps detect cycles (invalid inputs). Always consider edge cases like prefix conflicts and disconnected components to build a robust solution.
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