Prefix Search Using Trie in DSA C++ - Time & Space Complexity

Time Complexity: Prefix Search Using Trie

O(m)

Understanding Time Complexity

We want to understand how fast we can find words that start with a given prefix using a Trie.

How does the search time change as the prefix length or number of words grows?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

bool startsWith(TrieNode* root, const std::string& prefix) {
    TrieNode* node = root;
    for (char c : prefix) {
        if (!node->children[c - 'a'])
            return false;
        node = node->children[c - 'a'];
    }
    return true;
}
    

This code checks if any word in the Trie starts with the given prefix by following nodes for each prefix character.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Loop over each character in the prefix string.
• How many times: Exactly once per character in the prefix (length m).

How Execution Grows With Input

The time grows directly with the length of the prefix we search for.

Input Size (prefix length m)	Approx. Operations
10	10 checks
100	100 checks
1000	1000 checks

Pattern observation: The time increases linearly as the prefix length increases.

Final Time Complexity

Time Complexity: O(m)

This means the search time depends only on the prefix length, not on the number of words stored.

Common Mistake

[X] Wrong: "The search time depends on the total number of words in the Trie."

[OK] Correct: The search only follows nodes for the prefix characters, so it depends on prefix length, not total words.

Interview Connect

Knowing how prefix search scales helps you explain why Tries are efficient for autocomplete and dictionary lookups.

Self-Check

"What if the Trie stored uppercase and lowercase letters separately? How would the time complexity change?"

Practice

1. What is the main purpose of using a Trie data structure in prefix search?

easy

A. To reverse strings quickly

B. To sort words alphabetically

C. To store words in a hash table

D. To efficiently find all words that start with a given prefix

Solution

1. Step 1: Understand Trie usage

   A Trie stores characters of words in a tree structure, allowing quick prefix lookups.

2. Step 2: Identify prefix search goal

   Prefix search means finding all words starting with a given prefix efficiently, which Trie supports well.

3. Final Answer:

   To efficiently find all words that start with a given prefix -> Option D

4. Quick Check:

   Trie = Prefix search efficiency [OK]

Hint: Trie is built for prefix-based fast searches [OK]

Common Mistakes:

• Confusing Trie with sorting algorithms
• Thinking Trie is a hash table
• Assuming Trie reverses strings

2. Which of the following is the correct way to declare a Trie node in C++?

easy

A. struct TrieNode { bool isEnd; TrieNode* children[26]; };

B. struct TrieNode { int value; TrieNode children; };

C. class TrieNode { bool isEnd; vector<int> children; };

D. struct TrieNode { bool isEnd; TrieNode children[]; };

Solution

1. Step 1: Check Trie node structure

   A Trie node typically has a boolean to mark word end and an array of pointers for children nodes.

2. Step 2: Match correct syntax

   struct TrieNode { bool isEnd; TrieNode* children[26]; }; correctly declares a boolean and an array of 26 pointers for English lowercase letters.

3. Final Answer:

   struct TrieNode { bool isEnd; TrieNode* children[26]; }; -> Option A

4. Quick Check:

   Trie node = bool + pointer array [OK]

Hint: Trie nodes have boolean + fixed-size pointer array [OK]

Common Mistakes:

• Using vector<int> instead of pointers
• Missing pointer (*) in children array
• Declaring children as single object, not array

3. Given the following code snippet, what will be the output after inserting words "cat", "car", "dog" and searching prefix "ca"?

Trie trie;
trie.insert("cat");
trie.insert("car");
trie.insert("dog");
vector<string> result = trie.searchPrefix("ca");
for (auto &word : result) std::cout << word << " ";

medium

A. cat car

B. dog

C. cat dog car

D. car cat dog

Solution

1. Step 1: Insert words into Trie

   Words "cat", "car", and "dog" are inserted. Trie stores them by characters.

2. Step 2: Search prefix "ca"

   Searching prefix "ca" returns all words starting with "ca", which are "cat" and "car".

3. Final Answer:

   cat car -> Option A

4. Quick Check:

   Prefix "ca" matches "cat" and "car" [OK]

Hint: Search returns words starting exactly with prefix [OK]

Common Mistakes:

• Including words without the prefix
• Assuming order is alphabetical by default
• Confusing prefix search with full word search

4. Identify the bug in this Trie insert function snippet:

void insert(string word) {
  TrieNode* node = root;
  for (char c : word) {
    int index = c - 'a';
    if (!node->children[index])
      node->children[index] = new TrieNode();
    node = node->children[index];
  }
  node->isEnd = false;
}

medium

A. Children array should be vector, not array

B. isEnd should be set to true after insertion

C. Index calculation is incorrect

D. Root node is not initialized

Solution

1. Step 1: Review isEnd assignment

   After inserting a word, the node representing last character should mark end of word by setting isEnd to true.

2. Step 2: Identify incorrect isEnd value

   Code sets isEnd to false, which is wrong and will cause search failures.

3. Final Answer:

   isEnd should be set to true after insertion -> Option B

4. Quick Check:

   Mark word end with isEnd = true [OK]

Hint: Set isEnd true at word end to mark complete word [OK]

Common Mistakes:

• Setting isEnd to false instead of true
• Forgetting to initialize root node
• Miscomputing index from character

5. You want to find all words in a Trie that start with prefix "app" and have length exactly 5. Which approach correctly applies prefix search with length filtering?

hard

A. Use a hash map instead of Trie for prefix and length search

B. Search prefix "app" and return all words regardless of length

C. Search prefix "app" in Trie, then collect words with length 5 only

D. Search all words and filter those starting with "app" and length 5

Solution

1. Step 1: Perform prefix search "app" in Trie

   Traverse Trie nodes for prefix "app" to reach subtree containing all matching words.

2. Step 2: Filter collected words by length 5

   From subtree, collect only words whose length equals 5 to meet the length condition.

3. Final Answer:

   Search prefix "app" in Trie, then collect words with length 5 only -> Option C

4. Quick Check:

   Prefix search + length filter = Search prefix "app" in Trie, then collect words with length 5 only [OK]

Hint: Filter length after prefix search for efficient results [OK]

Common Mistakes:

• Returning all prefix matches without length check
• Filtering all words instead of prefix subtree
• Using hash map losing prefix search efficiency