[Hard] 269. Alien Dictionary

There is a new alien language that uses the English alphabet. However, the order among the letters is unknown to you.

You are given a list of strings words from the alien language’s dictionary, where the strings are sorted lexicographically by the rules of this new language.

Return a string of the unique letters in the new alien language sorted in lexicographically increasing order by the new language’s rules. If there is no solution, return "". If there are multiple solutions, return any of them.

Examples

Example 1:

Input: words = ["wrt","wrf","er","ett","rftt"]
Output: "wertf"

Example 2:

Input: words = ["z","x"]
Output: "zx"

Example 3:

Input: words = ["z","x","z"]
Output: ""
Explanation: The order is invalid, so return "".

Constraints

1 <= words.length <= 100
1 <= words[i].length <= 100
words[i] consists of only lowercase English letters.

Thinking Process

Graph Construction: Compare adjacent words to extract ordering constraints

Model entities as nodes and relationships as edges.
Pick traversal (BFS/DFS) or shortest-path (Dijkstra) based on weights.
Union-Find helps when connectivity updates are frequent.

Common Approaches

Typical techniques for this pattern:

Approach	Time	Space	Notes
Queue BFS (this problem)	O(n)	O(n)	Shortest path in unweighted graphs
Multi-source BFS	O(n)	O(n)	Start from all sources simultaneously
0-1 BFS / deque	O(n)	O(n)	Weights 0 or 1
Level-order BFS	O(n)	O(w)	Process by depth/layer

Solution

Solution: Topological Sort with BFS (Kahn’s Algorithm)

from collections import defaultdict, deque

class Solution:
    def alienOrder(self, words):
        # Step 1: build graph
        adj = defaultdict(set)
        indegree = {c: 0 for word in words for c in word}

        # Step 2: build edges from adjacent words
        for i in range(len(words) - 1):
            w1, w2 = words[i], words[i + 1]
            minLen = min(len(w1), len(w2))

            # invalid case: prefix issue
            if len(w1) > len(w2) and w1[:minLen] == w2[:minLen]:
                return ""

            for j in range(minLen):
                if w1[j] != w2[j]:
                    if w2[j] not in adj[w1[j]]:
                        adj[w1[j]].add(w2[j])
                        indegree[w2[j]] += 1
                    break

        # Step 3: BFS topo sort
        q = deque([c for c in indegree if indegree[c] == 0])
        result = []

        while q:
            node = q.popleft()
            result.append(node)

            for nei in adj[node]:
                indegree[nei] -= 1
                if indegree[nei] == 0:
                    q.append(nei)

        # Step 4: cycle check
        if len(result) != len(indegree):
            return ""

        return "".join(result)

Solution Explanation

Approach: Queue BFS (this problem)

Key idea: 1. Graph Construction: Compare adjacent words to extract ordering constraints

How the code works:

Graph Construction: Compare adjacent words to extract ordering constraints
- Model entities as nodes and relationships as edges.
- Pick traversal (BFS/DFS) or shortest-path (Dijkstra) based on weights.
- Union-Find helps when connectivity updates are frequent.

Walkthrough — input words = ["wrt","wrf","er","ett","rftt"], expected output "wertf":

Initialize variables from the problem setup.
Apply the main loop / recursion until the condition is met.
Confirm the result matches the expected output.

Algorithm Explanation:

Graph Initialization (Lines 7-13):
- Create adjacency list adj for all characters
- Initialize all characters from words (even if no edges)
Build Graph from Word Comparisons (Lines 15-30):
- Compare adjacent words words[i] and words[i+1]
- Find first differing character at position j
- Create edge: adj[w1[j]] → w2[j] (w1[j] comes before w2[j])
- Increment indegree: inDegree[w2[j] - 'a']++
- Invalid order check: If w1 is prefix of w2 but w1.size() > w2.size(), return ""
- Break after first difference: Only first differing character gives ordering
Topological Sort (BFS) (Lines 32-45):
- Find sources: Characters with inDegree = 0 (no prerequisites)
- Process queue:
  - Remove character from queue
  - Add to result
  - Reduce indegrees of all neighbors
  - Add neighbors to queue if indegree becomes 0
Validation (Lines 47-51):
- Check completeness: If rtn.size() == adj.size(), all characters processed (valid DAG)
- Otherwise: Cycle exists or invalid order → return ""

Why This Works:

Graph Model: Characters are nodes, ordering relationships are directed edges
Topological Sort: Finds valid ordering that respects all constraints
Cycle Detection: If not all nodes processed, there’s a cycle (contradictory orderings)
Invalid Prefix: If longer word comes before shorter word with same prefix, order is invalid

Example Walkthrough:

Input: words = ["wrt","wrf","er","ett","rftt"]

Step 1: Build Graph

Compare "wrt" and "wrf":
  First difference at index 2: 't' != 'f'
  Edge: t → f
  inDegree['f'] = 1

Compare "wrf" and "er":
  First difference at index 0: 'w' != 'e'
  Edge: w → e
  inDegree['e'] = 1

Compare "er" and "ett":
  First difference at index 1: 'r' != 't'
  Edge: r → t
  inDegree['t'] = 1

Compare "ett" and "rftt":
  First difference at index 0: 'e' != 'r'
  Edge: e → r
  inDegree['r'] = 1

Graph:
  w → e → r → t → f
  t → f

Indegrees:
  w: 0, e: 1, r: 1, t: 1, f: 2

Step 2: Topological Sort

Initial queue: [w] (indegree = 0)

Process 'w':
  rtn = "w"
  Neighbors: [e]
  inDegree['e'] = 0 → add 'e' to queue
  Queue: [e]

Process 'e':
  rtn = "we"
  Neighbors: [r]
  inDegree['r'] = 0 → add 'r' to queue
  Queue: [r]

Process 'r':
  rtn = "wer"
  Neighbors: [t]
  inDegree['t'] = 0 → add 't' to queue
  Queue: [t]

Process 't':
  rtn = "wert"
  Neighbors: [f]
  inDegree['f'] = 1 → not ready
  Queue: []

Wait, we need to check all characters. Let me recalculate:

Actually, after processing 't':
  inDegree['f'] = 2 - 1 = 1 (still not 0)
  
But we've processed all nodes with indegree 0. Let me check the graph again.

Actually, the issue is that 'f' has indegree 2 (from 't' and from the first comparison).
After processing 't', inDegree['f'] becomes 1, not 0.

Let me trace more carefully:
- w → e (inDegree['e'] = 1)
- e → r (inDegree['r'] = 1)  
- r → t (inDegree['t'] = 1)
- t → f (inDegree['f'] = 1, but we also have t → f from first comparison)

Wait, the code checks `!adj[w1[j]].contains(w2[j])` to avoid duplicate edges.
So if we already have t → f, we don't add it again.

Let me recalculate:
- "wrt" vs "wrf": t → f, inDegree['f'] = 1
- "wrf" vs "er": w → e, inDegree['e'] = 1
- "er" vs "ett": r → t, inDegree['t'] = 1
- "ett" vs "rftt": e → r, inDegree['r'] = 1

So:
- w: indegree 0
- e: indegree 1 (from w)
- r: indegree 1 (from e)
- t: indegree 1 (from r)
- f: indegree 1 (from t)

Processing:
1. Process w → e indegree becomes 0
2. Process e → r indegree becomes 0
3. Process r → t indegree becomes 0
4. Process t → f indegree becomes 0
5. Process f

Result: "wertf" ✓

Complexity Analysis:

Time Complexity: O(C + E) where C is number of unique characters, E is number of edges
- Building graph: O(N × L) where N is number of words, L is average word length
- Topological sort: O(C + E)
- Overall: O(N × L + C + E) = O(N × L) since E ≤ C²
Space Complexity: O(C + E)
- Adjacency list: O(C + E)
- Indegree array: O(C)
- Queue: O(C)
  Common Mistakes

Single word: words = ["abc"] → return “abc” (any order)
Invalid prefix: words = ["abc", "ab"] → return “” (invalid)
Cycle: words = ["a", "b", "a"] → return “” (cycle)
No constraints: words = ["a", "b"] → return “ab” or “ba” (both valid)
All same prefix: words = ["ab", "ac", "ad"] → extract ordering from first difference
Not checking invalid prefix: Forgetting to check if longer word comes before shorter
Duplicate edges: Not checking if edge already exists before adding
Missing characters: Not initializing all characters in adjacency list
Wrong indegree calculation: Incrementing indegree for wrong character
Not validating result: Not checking if all characters were processed

LC 207: Course Schedule - Topological sort to detect cycles
LC 210: Course Schedule II - Topological sort to find ordering
LC 269: Alien Dictionary - This problem
LC 444: Sequence Reconstruction - Verify unique topological order

Key Takeaways

Graph Construction: Compare adjacent words to extract ordering constraints
First Difference Rule: Only the first differing character gives ordering information
Invalid Prefix: Longer word cannot come before shorter word with same prefix
Topological Sort: Use BFS (Kahn’s algorithm) to find valid ordering
Cycle Detection: If not all nodes processed, cycle exists (invalid order)

References

LC 269: Alien Dictionary on LeetCode
LeetCode Discuss — LC 269: Alien Dictionary
LeetCode Editorial (may require premium)

Template Reference

Graph

Robina Li

[Hard] 269. Alien Dictionary

Examples

Constraints

Thinking Process

Common Approaches

Solution

Solution: Topological Sort with BFS (Kahn’s Algorithm)

Solution Explanation

Algorithm Explanation:

Why This Works:

Example Walkthrough:

Complexity Analysis:

Common Mistakes

Key Takeaways

References

Template Reference

Related Posts

Recent Posts

Examples

Constraints

Thinking Process

Common Approaches

Solution

Solution: Topological Sort with BFS (Kahn’s Algorithm)

Solution Explanation

Algorithm Explanation:

Why This Works:

Example Walkthrough:

Complexity Analysis:

Common Mistakes

Related Problems

Key Takeaways

References

Template Reference

Related Posts

Recent Posts