Longest Word in Dictionary Using Trie in DSA C - Time & Space Complexity

Time Complexity: Longest Word in Dictionary Using Trie

O(n * m)

Understanding Time Complexity

We want to understand how the time needed grows when finding the longest word using a Trie.

How does the work increase as the list of words gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

// Insert words into Trie
for (int i = 0; i < n; i++) {
    insert(root, words[i]);
}
// Find longest word with all prefixes present
char* longestWord = findLongestWord(root);
    

This code inserts all words into a Trie and then finds the longest word where every prefix is also a word.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Inserting each word character by character into the Trie.
• How many times: For each of the n words, each with average length m, we do m steps.
• Secondary operation: Traversing the Trie to find the longest valid word, visiting nodes up to total characters inserted.

How Execution Grows With Input

As the number of words (n) or their average length (m) grows, the work grows roughly by n times m.

Input Size (n)	Approx. Operations
10 words, avg length 5	About 50 insert steps + traversal steps
100 words, avg length 5	About 500 insert steps + traversal steps
1000 words, avg length 5	About 5000 insert steps + traversal steps

Pattern observation: The total steps increase roughly linearly with the total number of characters in all words.

Final Time Complexity

Time Complexity: O(n * m)

This means the time grows linearly with the total number of characters in all words combined.

Common Mistake

[X] Wrong: "The search for the longest word is just O(n) because we only check each word once."

[OK] Correct: Actually, inserting each word takes time proportional to its length, and the Trie traversal also depends on total characters, so length matters too.

Interview Connect

Understanding how Trie operations scale helps you explain efficient word searches and prefix checks clearly in interviews.

Self-Check

"What if the words were sorted by length before insertion? How would that affect the time complexity?"

Practice

1. What is the main purpose of using a Trie data structure when finding the longest word in a dictionary?

easy

A. To sort words alphabetically

B. To count the total number of words

C. To efficiently store and search words by their prefixes

D. To reverse the words in the dictionary

Solution

1. Step 1: Understand Trie usage

   A Trie stores words by breaking them into characters and linking nodes for prefixes.

2. Step 2: Relate to longest word search

   This structure allows quick prefix checks, which helps find the longest word built from other words.

3. Final Answer:

   To efficiently store and search words by their prefixes -> Option C

4. Quick Check:

   Trie = prefix search [OK]

Hint: Trie is best for prefix-based word searches [OK]

Common Mistakes:

• Thinking Trie sorts words alphabetically
• Confusing Trie with simple arrays or lists
• Assuming Trie counts words only

2. Which of the following is the correct way to declare a Trie node structure in C for lowercase English letters?

easy

A. typedef struct TrieNode { struct TrieNode* children[26]; int isEndOfWord; } TrieNode;

B. typedef struct TrieNode { struct TrieNode* children[52]; int isEndOfWord; } TrieNode;

C. typedef struct TrieNode { struct TrieNode* children[10]; int isEndOfWord; } TrieNode;

D. typedef struct TrieNode { struct TrieNode* children; int isEndOfWord; } TrieNode;

Solution

1. Step 1: Identify children array size

   Lowercase English letters are 26, so children array size must be 26.

2. Step 2: Check structure fields

   Each node has an array of pointers and a flag for word end, matching typedef struct TrieNode { struct TrieNode* children[26]; int isEndOfWord; } TrieNode;.

3. Final Answer:

   typedef struct TrieNode { struct TrieNode* children[26]; int isEndOfWord; } TrieNode; -> Option A

4. Quick Check:

   26 children for a-z [OK]

Hint: Use 26 children for lowercase letters [OK]

Common Mistakes:

• Using 52 for uppercase + lowercase without need
• Using 10 children for digits only
• Not using array for children pointers

3. Given the following code snippet inserting words into a Trie and searching for the longest word built one character at a time, what will be the output?

char* words[] = {"a", "ap", "app", "appl", "apple", "apply"};
// Assume insert and search functions are correct
// Function finds longest word where all prefixes exist
printf("%s", longestWord(words, 6));

medium

A. apple

B. apply

C. appl

D. app

Solution

1. Step 1: Check all prefixes for each word

   Words like "apple" and "apply" have all prefixes present: a, ap, app, appl.

2. Step 2: Compare lengths of valid words

   "apple" and "apply" both have length 5, but "apple" comes first lexicographically.

3. Final Answer:

   apple -> Option A

4. Quick Check:

   Longest valid word = apple [OK]

Hint: Check all prefixes exist, pick longest lex order first [OK]

Common Mistakes:

• Choosing 'apply' ignoring lex order
• Picking shorter words like 'app'
• Not verifying all prefixes exist

4. The following code tries to find the longest word in a dictionary using a Trie but always returns an empty string. What is the bug?

void insert(TrieNode* root, char* word) {
    TrieNode* node = root;
    for (int i = 0; word[i]; i++) {
        int index = word[i] - 'a';
        if (!node->children[index])
            node->children[index] = createNode();
        node = node->children[index];
    }
    node->isEndOfWord = 1;
}

char* longestWord(char** words, int n) {
    TrieNode* root = createNode();
    for (int i = 0; i < n; i++)
        insert(root, words[i]);
    // Missing code to find longest word
    return "";
}

medium

A. The children array size is incorrect

B. The longestWord function does not search the Trie for the longest word

C. The insert function does not mark end of word correctly

D. The root node is not initialized

Solution

1. Step 1: Analyze insert function

   Insert correctly adds words and marks end of word, so no issue here.

2. Step 2: Check longestWord function

   It inserts words but returns empty string without searching Trie for longest word.

3. Final Answer:

   The longestWord function does not search the Trie for the longest word -> Option B

4. Quick Check:

   Missing search logic causes empty output [OK]

Hint: Check if search logic exists after insertion [OK]

Common Mistakes:

• Assuming insert alone finds longest word
• Ignoring missing traversal in longestWord
• Thinking root node is uninitialized

5. You want to find the longest word in a dictionary such that every prefix of the word is also a word in the dictionary. Which approach using a Trie is best to solve this efficiently?

hard

A. Use BFS on Trie without checking end of word flags

B. Sort words by length, then check each word's prefixes in a hash set

C. Insert words into Trie and count total nodes to find longest word

D. Insert all words into Trie, then do DFS checking only nodes marking end of words to build longest word

Solution

1. Step 1: Insert all words into Trie

   Build Trie with all dictionary words, marking end of word nodes.

2. Step 2: Use DFS to find longest word

   Traverse Trie depth-first, only continuing on nodes that mark end of words to ensure all prefixes exist.

3. Step 3: Track longest word during traversal

   Keep updating longest word found by length and lex order during DFS.

4. Final Answer:

   Insert all words into Trie, then do DFS checking only nodes marking end of words to build longest word -> Option D

5. Quick Check:

   Trie + DFS + endOfWord check = longest word [OK]

Hint: Use DFS on Trie nodes with endOfWord true only [OK]

Common Mistakes:

• Ignoring prefix checks during traversal
• Using BFS without endOfWord validation
• Relying on sorting without Trie